L-relations and Galois triangles Basic notions Adjoint product - - PowerPoint PPT Presentation

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

L-relations and Galois triangles

M.Emilia Della Stella 1 Cosimo Guido 2

1Department of Mathematics-University of Trento 2Department of Mathematics-University of Salento

May 20, 2011

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Preliminaries Galois connections Extended-order algebras

Basic notions Adjoint product Symmetry Commutativity and associativity

L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ An adjuction also called isotonic Galois connection between

two posets L and M, denoted by f Ú g, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: ❇ ❼ ➁ ✔ ❼ ➁ ❇ ➛ ❃ ❃

▲

• ✆

✂

• ✂
• ❇

❼ ➁ ✔ ❇ ❼ ➁ ➛ ❃ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ An adjuction also called isotonic Galois connection between

two posets L and M, denoted by f Ú g, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: x ❇ g❼y➁ ✔ f ❼x➁ ❇ y, ➛x ❃ L,y ❃ M.

▲

• ✆

✂

• ✂
• ❇

❼ ➁ ✔ ❇ ❼ ➁ ➛ ❃ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ An adjuction also called isotonic Galois connection between

two posets L and M, denoted by f Ú g, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: x ❇ g❼y➁ ✔ f ❼x➁ ❇ y, ➛x ❃ L,y ❃ M. The map f is called left adjoint of g and g right adjoint of f .

▲

• ✆

✂

• ✂
• ❇

❼ ➁ ✔ ❇ ❼ ➁ ➛ ❃ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ An adjuction also called isotonic Galois connection between

two posets L and M, denoted by f Ú g, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: x ❇ g❼y➁ ✔ f ❼x➁ ❇ y, ➛x ❃ L,y ❃ M. The map f is called left adjoint of g and g right adjoint of f .

▲ An (antitonic) Galois connection between two posets L and

M, denoted by f ,g✆, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: ❇ ❼ ➁ ✔ ❇ ❼ ➁ ➛ ❃ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ An adjuction also called isotonic Galois connection between

two posets L and M, denoted by f Ú g, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: x ❇ g❼y➁ ✔ f ❼x➁ ❇ y, ➛x ❃ L,y ❃ M. The map f is called left adjoint of g and g right adjoint of f .

▲ An (antitonic) Galois connection between two posets L and

M, denoted by f ,g✆, is a pair of maps f ✂ L M and g ✂ M L satisfying the following condition: x ❇ g❼y➁ ✔ y ❇ f ❼x➁, ➛x ❃ L,y ❃ M.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ ▲

▲

✂

• ✂
• ③
• ❼ ➁ ✝➌ ❃ ❙

❼ ➁ ❇ ➑

▲

✂

• ✂
• ③

❼ ➁ ☎➌ ❃ ❙ ❇ ❼ ➁➑

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ f preserves existing sups; ▲

▲

✂

• ✂
• ③
• ❼ ➁ ✝➌ ❃ ❙

❼ ➁ ❇ ➑

▲

✂

• ✂
• ③

❼ ➁ ☎➌ ❃ ❙ ❇ ❼ ➁➑

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ f preserves existing sups; ▲ g preserves existing infs.

▲

✂

• ✂
• ③
• ❼ ➁ ✝➌ ❃ ❙

❼ ➁ ❇ ➑

▲

✂

• ✂
• ③

❼ ➁ ☎➌ ❃ ❙ ❇ ❼ ➁➑

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ f preserves existing sups; ▲ g preserves existing infs.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

map that preserves sups. ✂

• ③
• ❼ ➁ ✝➌ ❃ ❙

❼ ➁ ❇ ➑

▲

✂

• ✂
• ③

❼ ➁ ☎➌ ❃ ❙ ❇ ❼ ➁➑

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ f preserves existing sups; ▲ g preserves existing infs.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

map that preserves sups.Then the function g ✂ M L, y ③ g❼y➁ ✝➌x ❃ L❙ f ❼x➁ ❇ y➑ is the unique right adjoint of f .

▲

✂

• ✂
• ③

❼ ➁ ☎➌ ❃ ❙ ❇ ❼ ➁➑

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ f preserves existing sups; ▲ g preserves existing infs.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

map that preserves sups.Then the function g ✂ M L, y ③ g❼y➁ ✝➌x ❃ L❙ f ❼x➁ ❇ y➑ is the unique right adjoint of f .

▲ Let L be a poset, M a complete lattice and let g ✂ M L be

a map that preserves infs. ✂

• ③

❼ ➁ ☎➌ ❃ ❙ ❇ ❼ ➁➑

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Remark

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. If f Ú g, then:

▲ f preserves existing sups; ▲ g preserves existing infs.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

map that preserves sups.Then the function g ✂ M L, y ③ g❼y➁ ✝➌x ❃ L❙ f ❼x➁ ❇ y➑ is the unique right adjoint of f .

▲ Let L be a poset, M a complete lattice and let g ✂ M L be

a map that preserves infs.Then the function f ✂ L M, x ③ f ❼x➁ ☎➌y ❃ M❙ x ❇ g❼y➁➑ is the unique left adjoint of g.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. Then

▲

✆

• ✆

▲

• ✆

❜ ❼✝ ➁ ☎ ❼ ➁

▲

✂

• ❼✝

➁ ☎ ❼ ➁ ✂

• ③
• ❼ ➁ ✝➌ ❃ ❙

❇ ❼ ➁➑

• ✆

5 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. Then

▲ g, f ✆ iff f , g✆ is; ▲

• ✆

❜ ❼✝ ➁ ☎ ❼ ➁

▲

✂

• ❼✝

➁ ☎ ❼ ➁ ✂

• ③
• ❼ ➁ ✝➌ ❃ ❙

❇ ❼ ➁➑

• ✆

5 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. Then

▲ g, f ✆ iff f , g✆ is; ▲ if f , g✆ and A ❜ L has a supremum, then f ❼✝ A➁ ☎ f ❼A➁.

▲

✂

• ❼✝

➁ ☎ ❼ ➁ ✂

• ③
• ❼ ➁ ✝➌ ❃ ❙

❇ ❼ ➁➑

• ✆

5 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. Then

▲ g, f ✆ iff f , g✆ is; ▲ if f , g✆ and A ❜ L has a supremum, then f ❼✝ A➁ ☎ f ❼A➁.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

function such that f ❼✝A➁ ☎f ❼A➁. ✂

• ③
• ❼ ➁ ✝➌ ❃ ❙

❇ ❼ ➁➑

• ✆

5 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. Then

▲ g, f ✆ iff f , g✆ is; ▲ if f , g✆ and A ❜ L has a supremum, then f ❼✝ A➁ ☎ f ❼A➁.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

function such that f ❼✝A➁ ☎f ❼A➁. ✂

• ③
• ❼ ➁ ✝➌ ❃ ❙

❇ ❼ ➁➑

• ✆

5 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ Let L and M be posets and let f ✂ L M and g ✂ M L be

• maps. Then

▲ g, f ✆ iff f , g✆ is; ▲ if f , g✆ and A ❜ L has a supremum, then f ❼✝ A➁ ☎ f ❼A➁.

▲ Let L be a complete lattice, M a poset and let f ✂ L M be a

function such that f ❼✝A➁ ☎f ❼A➁.Then the function g ✂ M L, y ③ g❼y➁ ✝➌x ❃ L❙ y ❇ f ❼x➁➑ is the unique function such that f ,g✆.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ C. Guido, P. Toto: Extended-order algebras, Journal of

Applied Logic, 6(4) (2008), 609-626.

▲ ▲

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ C. Guido, P. Toto: Extended-order algebras, Journal of

Applied Logic, 6(4) (2008), 609-626.

▲ H. Rasiowa: An Algebraic Approach to Non-Classical Logics,

Studies in Logics and the Foundations of Mathematics, vol.78, North-Holland, Amsterdam, 1974.

▲

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ C. Guido, P. Toto: Extended-order algebras, Journal of

Applied Logic, 6(4) (2008), 609-626.

▲ H. Rasiowa: An Algebraic Approach to Non-Classical Logics,

Studies in Logics and the Foundations of Mathematics, vol.78, North-Holland, Amsterdam, 1974.

▲ M.E.D.S., C. Guido: Associativity, commutativity and

symmetry in residuated structures, (submitted).

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let L be a non-empty set, ✂ L ✕ L L a binary operation and ➋ a fixed element of L. The triple L ❼L,,➋➁ is a weak extended-order algebra, shortly w-eo algebra, if

▲ ❼

➁ ➋ ➋

▲ ❼

➁

• ➋

▲ ❼

➁

• ➋
• ➋ ✟
• ▲ ❼

➁

• ➋
• ➋ ✟
• ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let L be a non-empty set, ✂ L ✕ L L a binary operation and ➋ a fixed element of L. The triple L ❼L,,➋➁ is a weak extended-order algebra, shortly w-eo algebra, if

▲ ❼o1➁ a ➋ ➋ (upper bound condition); ▲ ❼

➁

• ➋

▲ ❼

➁

• ➋
• ➋ ✟
• ▲ ❼

➁

• ➋
• ➋ ✟
• ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let L be a non-empty set, ✂ L ✕ L L a binary operation and ➋ a fixed element of L. The triple L ❼L,,➋➁ is a weak extended-order algebra, shortly w-eo algebra, if

▲ ❼o1➁ a ➋ ➋ (upper bound condition); ▲ ❼o2➁ a a ➋ (reflexivity condition); ▲ ❼

➁

• ➋
• ➋ ✟
• ▲ ❼

➁

• ➋
• ➋ ✟
• ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let L be a non-empty set, ✂ L ✕ L L a binary operation and ➋ a fixed element of L. The triple L ❼L,,➋➁ is a weak extended-order algebra, shortly w-eo algebra, if

▲ ❼o1➁ a ➋ ➋ (upper bound condition); ▲ ❼o2➁ a a ➋ (reflexivity condition); ▲ ❼o3➁ a b ➋ and b a ➋ ✟ a b (antisymmetry

condition);

▲ ❼

➁

• ➋
• ➋ ✟
• ➋

7 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let L be a non-empty set, ✂ L ✕ L L a binary operation and ➋ a fixed element of L. The triple L ❼L,,➋➁ is a weak extended-order algebra, shortly w-eo algebra, if

▲ ❼o1➁ a ➋ ➋ (upper bound condition); ▲ ❼o2➁ a a ➋ (reflexivity condition); ▲ ❼o3➁ a b ➋ and b a ➋ ✟ a b (antisymmetry

condition);

▲ ❼o4➁ a b ➋ and b c ➋ ✟ a c ➋ (weak transitivity

condition).

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲

❇

• ❇
• ➋

➋ ❼ ❇➁

▲

❼ ❇➁ ➋ ✂ ✕

• ❇
• ➋ ✔

❇ ❼ ➋➁ ❼ ➋➁ ❼ ➁ ❼ ➁ ❼ ➁

▲ ❼

➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

▲ ❼ ➐➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ The relation ❇ determined by the operation , by means of

the equivalence a ❇ b iff a b ➋ is an order relation in L. Moreover ➋ is a greatest element in ❼L,❇➁.

▲

❼ ❇➁ ➋ ✂ ✕

• ❇
• ➋ ✔

❇ ❼ ➋➁ ❼ ➋➁ ❼ ➁ ❼ ➁ ❼ ➁

▲ ❼

➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

▲ ❼ ➐➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ The relation ❇ determined by the operation , by means of

the equivalence a ❇ b iff a b ➋ is an order relation in L. Moreover ➋ is a greatest element in ❼L,❇➁.

▲ Conversely, if ❼L,❇➁ is a poset with a greatest element ➋ and

✂ L ✕ L L extends ❇, i.e. a b ➋ ✔ a ❇ b, then ❼L,,➋➁ is a w-eo algebra. ❼ ➋➁ ❼ ➁ ❼ ➁ ❼ ➁

▲ ❼

➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

▲ ❼ ➐➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ The relation ❇ determined by the operation , by means of

the equivalence a ❇ b iff a b ➋ is an order relation in L. Moreover ➋ is a greatest element in ❼L,❇➁.

▲ Conversely, if ❼L,❇➁ is a poset with a greatest element ➋ and

✂ L ✕ L L extends ❇, i.e. a b ➋ ✔ a ❇ b, then ❼L,,➋➁ is a w-eo algebra. ❼ ➋➁ ❼ ➁ ❼ ➁ ❼ ➁

▲ ❼

➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

▲ ❼ ➐➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ The relation ❇ determined by the operation , by means of

the equivalence a ❇ b iff a b ➋ is an order relation in L. Moreover ➋ is a greatest element in ❼L,❇➁.

▲ Conversely, if ❼L,❇➁ is a poset with a greatest element ➋ and

✂ L ✕ L L extends ❇, i.e. a b ➋ ✔ a ❇ b, then ❼L,,➋➁ is a w-eo algebra.

Definition

❼L,,➋➁ is an extended-order algebra, shortly eo algebra, if it satisfies the axioms ❼o1➁, ❼o2➁, ❼o3➁ and

▲ ❼

➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

▲ ❼ ➐➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ The relation ❇ determined by the operation , by means of

the equivalence a ❇ b iff a b ➋ is an order relation in L. Moreover ➋ is a greatest element in ❼L,❇➁.

▲ Conversely, if ❼L,❇➁ is a poset with a greatest element ➋ and

✂ L ✕ L L extends ❇, i.e. a b ➋ ✔ a ❇ b, then ❼L,,➋➁ is a w-eo algebra.

Definition

❼L,,➋➁ is an extended-order algebra, shortly eo algebra, if it satisfies the axioms ❼o1➁, ❼o2➁, ❼o3➁ and

▲ ❼o5➁ a b ➋ ✟ ❼c a➁ ❼c b➁ ➋(weak isotonic

condition in the second variable);

▲ ❼ ➐➁

• ➋ ✟ ❼ ➁ ❼ ➁ ➋

8 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ The relation ❇ determined by the operation , by means of

the equivalence a ❇ b iff a b ➋ is an order relation in L. Moreover ➋ is a greatest element in ❼L,❇➁.

▲ Conversely, if ❼L,❇➁ is a poset with a greatest element ➋ and

✂ L ✕ L L extends ❇, i.e. a b ➋ ✔ a ❇ b, then ❼L,,➋➁ is a w-eo algebra.

Definition

❼L,,➋➁ is an extended-order algebra, shortly eo algebra, if it satisfies the axioms ❼o1➁, ❼o2➁, ❼o3➁ and

▲ ❼o5➁ a b ➋ ✟ ❼c a➁ ❼c b➁ ➋(weak isotonic

condition in the second variable);

▲ ❼o➐ 5➁ a b ➋ ✟ ❼b c➁ ❼a c➁ ➋(weak antitonic

condition in the first variable).

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Any one of the algebras defined above is said to be complete if L with the order induced by is a complete lattice.

▲ ▲

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Any one of the algebras defined above is said to be complete if L with the order induced by is a complete lattice. From now, we consider only complete structures and we denote them with the obvious notation (w-)ceo algebras.

▲ ▲

9 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Any one of the algebras defined above is said to be complete if L with the order induced by is a complete lattice. From now, we consider only complete structures and we denote them with the obvious notation (w-)ceo algebras.

Remark

The completeness requirement is

▲ ▲

9 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Any one of the algebras defined above is said to be complete if L with the order induced by is a complete lattice. From now, we consider only complete structures and we denote them with the obvious notation (w-)ceo algebras.

Remark

The completeness requirement is

▲ not restrictive for eo algebras; ▲

9 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Any one of the algebras defined above is said to be complete if L with the order induced by is a complete lattice. From now, we consider only complete structures and we denote them with the obvious notation (w-)ceo algebras.

Remark

The completeness requirement is

▲ not restrictive for eo algebras; ▲ restrictive for w-eo algebras.

9 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼

➋➁ ❼ ➁ ☎ ☎❼ ➁

▲ ❼

➋➁ ❼ ➁ ❼✝ ➁ ☎❼ ➁

▲ ❼

➋➁ ❼ ➁ ✝ ☎ ☎❼

• ➁

▲ ❼

➁ ✟ ❼ ➁

▲ ❼

➁ ✟ ❼

➐➁ ▲ ❼

➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼

➐➁

10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼L,,➋➁ is right-distributive if it satisfies

❼dr➁ a ☎B ☎❼a B➁.

▲ ❼

➋➁ ❼ ➁ ❼✝ ➁ ☎❼ ➁

▲ ❼

➋➁ ❼ ➁ ✝ ☎ ☎❼

• ➁

▲ ❼

➁ ✟ ❼ ➁

▲ ❼

➁ ✟ ❼

➐➁ ▲ ❼

➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼

➐➁

10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼L,,➋➁ is right-distributive if it satisfies

❼dr➁ a ☎B ☎❼a B➁.

▲ ❼L,,➋➁ is left-distributive if it satisfies

❼dl➁ ❼✝A➁ b ☎❼A b➁.

▲ ❼

➋➁ ❼ ➁ ✝ ☎ ☎❼

• ➁

▲ ❼

➁ ✟ ❼ ➁

▲ ❼

➁ ✟ ❼

➐➁ ▲ ❼

➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼

➐➁

10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼L,,➋➁ is right-distributive if it satisfies

❼dr➁ a ☎B ☎❼a B➁.

▲ ❼L,,➋➁ is left-distributive if it satisfies

❼dl➁ ❼✝A➁ b ☎❼A b➁.

▲ ❼L,,➋➁ is distributive if it satisfies

❼d➁ ✝A ☎B ☎❼A B➁.

▲ ❼

➁ ✟ ❼ ➁

▲ ❼

➁ ✟ ❼

➐➁ ▲ ❼

➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼

➐➁

10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼L,,➋➁ is right-distributive if it satisfies

❼dr➁ a ☎B ☎❼a B➁.

▲ ❼L,,➋➁ is left-distributive if it satisfies

❼dl➁ ❼✝A➁ b ☎❼A b➁.

▲ ❼L,,➋➁ is distributive if it satisfies

❼d➁ ✝A ☎B ☎❼A B➁.

Remark

▲ ❼dr➁ ✟ ❼o5➁; ▲ ❼

➁ ✟ ❼

➐➁ ▲ ❼

➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼

➐➁

10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼L,,➋➁ is right-distributive if it satisfies

❼dr➁ a ☎B ☎❼a B➁.

▲ ❼L,,➋➁ is left-distributive if it satisfies

❼dl➁ ❼✝A➁ b ☎❼A b➁.

▲ ❼L,,➋➁ is distributive if it satisfies

❼d➁ ✝A ☎B ☎❼A B➁.

Remark

▲ ❼dr➁ ✟ ❼o5➁; ▲ ❼dl➁ ✟ ❼o➐ 5➁; ▲ ❼

➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼

➐➁

10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a w-ceo algebra.

▲ ❼L,,➋➁ is right-distributive if it satisfies

❼dr➁ a ☎B ☎❼a B➁.

▲ ❼L,,➋➁ is left-distributive if it satisfies

❼dl➁ ❼✝A➁ b ☎❼A b➁.

▲ ❼L,,➋➁ is distributive if it satisfies

❼d➁ ✝A ☎B ☎❼A B➁.

Remark

▲ ❼dr➁ ✟ ❼o5➁; ▲ ❼dl➁ ✟ ❼o➐ 5➁; ▲ ❼dr➁+❼dl➁ ✔ ❼d➁ ✟ ❼o5➁+❼o➐ 5➁.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by ❛ ☎➌ ❃ ❙ ❇ ➑

▲ ❛

• ❛

❇ ✔ ❇

• ▲

✂

• ✭

❼ ➁

• ☎

❼ ➁

▲

❛ ➋

▲

❛ ➊ ➊ ❛ ➊

▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

▲ ❛

• ❛

❇ ✔ ❇

• ▲

✂

• ✭

❼ ➁

• ☎

❼ ➁

▲

❛ ➋

▲

❛ ➊ ➊ ❛ ➊

▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲

✂

• ✭

❼ ➁

• ☎

❼ ➁

▲

❛ ➋

▲

❛ ➊ ➊ ❛ ➊

▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲ The above Definition is justified by adjunction applied to the

function ga ✂ L L, y ✭ ga❼y➁ a y that preserves ☎, because the condition ❼dr➁ is assumed on L;

▲

❛ ➋

▲

❛ ➊ ➊ ❛ ➊

▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲ The above Definition is justified by adjunction applied to the

function ga ✂ L L, y ✭ ga❼y➁ a y that preserves ☎, because the condition ❼dr➁ is assumed on L;

Proposition

▲

❛ ➋

▲

❛ ➊ ➊ ❛ ➊

▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲ The above Definition is justified by adjunction applied to the

function ga ✂ L L, y ✭ ga❼y➁ a y that preserves ☎, because the condition ❼dr➁ is assumed on L;

Proposition

▲ a ❛ ➋ a; ▲

❛ ➊ ➊ ❛ ➊

▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲ The above Definition is justified by adjunction applied to the

function ga ✂ L L, y ✭ ga❼y➁ a y that preserves ☎, because the condition ❼dr➁ is assumed on L;

Proposition

▲ a ❛ ➋ a; ▲ a ❛ ➊ ➊ ❛ a ➊; ▲

❛ ❼✝ ➁ ✝❼ ❛ ➁

▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲ The above Definition is justified by adjunction applied to the

function ga ✂ L L, y ✭ ga❼y➁ a y that preserves ☎, because the condition ❼dr➁ is assumed on L;

Proposition

▲ a ❛ ➋ a; ▲ a ❛ ➊ ➊ ❛ a ➊; ▲ a ❛ ❼✝B➁ ✝❼a ❛ B➁; ▲

❛ ① ❛ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The adjoint product of L is the operation ❛ ✂ L ✕ L L defined by a ❛ b ☎➌t ❃ L❙ b ❇ a t➑.

Remark

▲ ❛ and form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x z. ▲ The above Definition is justified by adjunction applied to the

function ga ✂ L L, y ✭ ga❼y➁ a y that preserves ☎, because the condition ❼dr➁ is assumed on L;

Proposition

▲ a ❛ ➋ a; ▲ a ❛ ➊ ➊ ❛ a ➊; ▲ a ❛ ❼✝B➁ ✝❼a ❛ B➁; ▲ a ❛ b ① b ❛ a and a ❛ ❼b ❛ c➁ ① ❼a ❛ b➁ ❛ c, in general.

11 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is symmetrical if

▲ ➜ ❼

➝ ➋➁

▲

❇ ➝ ✔ ❇

• ➝

▲

• ➝

▲ ❼

➝ ➋➁ ❼ ➋➁

12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is symmetrical if

▲ ➜ ❼L,➝,➋➁ w-ceo algebra, with the same induced order; ▲

❇ ➝ ✔ ❇

• ➝

▲

• ➝

▲ ❼

➝ ➋➁ ❼ ➋➁

12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is symmetrical if

▲ ➜ ❼L,➝,➋➁ w-ceo algebra, with the same induced order; ▲ y ❇ x ➝ z ✔ x ❇ y z (i.e.[,➝]). ▲

• ➝

▲ ❼

➝ ➋➁ ❼ ➋➁

12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is symmetrical if

▲ ➜ ❼L,➝,➋➁ w-ceo algebra, with the same induced order; ▲ y ❇ x ➝ z ✔ x ❇ y z (i.e.[,➝]).

Remark

▲

• ➝

▲ ❼

➝ ➋➁ ❼ ➋➁

12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is symmetrical if

▲ ➜ ❼L,➝,➋➁ w-ceo algebra, with the same induced order; ▲ y ❇ x ➝ z ✔ x ❇ y z (i.e.[,➝]).

Remark

▲ Since and ➝ form a Galois pair, each of them is uniquely

determined by the other one.

▲ ❼

➝ ➋➁ ❼ ➋➁

12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is symmetrical if

▲ ➜ ❼L,➝,➋➁ w-ceo algebra, with the same induced order; ▲ y ❇ x ➝ z ✔ x ❇ y z (i.e.[,➝]).

Remark

▲ Since and ➝ form a Galois pair, each of them is uniquely

determined by the other one.

▲ ❼L,➝,➋➁ is symmetrical iff ❼L,,➋➁ is symmetrical.

12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼

➋➁

▲ ❼

➝ ➋➁

▲ ➋ ❛

• ▲ ❼✝

➁ ❛ ✝❼ ❛ ➁

▲

❛ ❼ ➝ ➋➁ ❛ ❛ ❛ ❛

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼

➝ ➋➁

▲ ➋ ❛

• ▲ ❼✝

➁ ❛ ✝❼ ❛ ➁

▲

❛ ❼ ➝ ➋➁ ❛ ❛ ❛ ❛

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛

• ▲ ❼✝

➁ ❛ ✝❼ ❛ ➁

▲

❛ ❼ ➝ ➋➁ ❛ ❛ ❛ ❛

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝

➁ ❛ ✝❼ ❛ ➁

▲

❛ ❼ ➝ ➋➁ ❛ ❛ ❛ ❛

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝B➁ ❛ a ✝❼B ❛ a➁. ▲

❛ ❼ ➝ ➋➁ ❛ ❛ ❛ ❛

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝B➁ ❛ a ✝❼B ❛ a➁.

Remark

▲

❛ ❼ ➝ ➋➁ ❛ ❛ ❛ ❛

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝B➁ ❛ a ✝❼B ❛ a➁.

Remark

▲ The adjoint product ˜

❛ of the cdeo algebra ❼L,➝,➋➁ is the

• pposite ❛op of ❛, i. e. a˜

❛b b ❛ a.

▲ ❛

➝ ❇ ➝ ✔ ❛ ❇

▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝B➁ ❛ a ✝❼B ❛ a➁.

Remark

▲ The adjoint product ˜

❛ of the cdeo algebra ❼L,➝,➋➁ is the

• pposite ❛op of ❛, i. e. a˜

❛b b ❛ a.

▲ ❛ and ➝ are related by the equivalence a ❇ b ➝ c ✔ a❛b ❇ c. ▲

➋ ❛ ①

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝B➁ ❛ a ✝❼B ❛ a➁.

Remark

▲ The adjoint product ˜

❛ of the cdeo algebra ❼L,➝,➋➁ is the

• pposite ❛op of ❛, i. e. a˜

❛b b ❛ a.

▲ ❛ and ➝ are related by the equivalence a ❇ b ➝ c ✔ a❛b ❇ c. ▲ The cdeo algebras need not to be symmetrical. In fact in the

cdeo algebras ➋ ❛ a ① a, in general.

▲

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Under right-distributivity and symmetry assumptions we have further properties.

Proposition

▲ ❼L,,➋➁ is a cdeo algebra; ▲ ❼L,➝,➋➁ is a cdeo algebra; ▲ ➋ ❛ a a; ▲ ❼✝B➁ ❛ a ✝❼B ❛ a➁.

Remark

▲ The adjoint product ˜

❛ of the cdeo algebra ❼L,➝,➋➁ is the

• pposite ❛op of ❛, i. e. a˜

❛b b ❛ a.

▲ ❛ and ➝ are related by the equivalence a ❇ b ➝ c ✔ a❛b ❇ c. ▲ The cdeo algebras need not to be symmetrical. In fact in the

cdeo algebras ➋ ❛ a ① a, in general.

▲ Symmetrical cdeo algebras are complete integral residuated

lattices, hence equivalent to pseudo BCK-algebras, without associativity.

13 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is commutative iff ❼c➁ a ❼b c➁ ➋ ✔ b ❼a c➁ ➋ (weak exchange condition). ❼ ➋➁

▲ ❼

➋➁

▲

❛

▲ ❼

➋➁ ➝

• ❛ ❛

14 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is commutative iff ❼c➁ a ❼b c➁ ➋ ✔ b ❼a c➁ ➋ (weak exchange condition).

Proposition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The followings are equivalent:

▲ ❼

➋➁

▲

❛

▲ ❼

➋➁ ➝

• ❛ ❛

14 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is commutative iff ❼c➁ a ❼b c➁ ➋ ✔ b ❼a c➁ ➋ (weak exchange condition).

Proposition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The followings are equivalent:

▲ ❼L,,➋➁ is commutative; ▲

❛

▲ ❼

➋➁ ➝

• ❛ ❛

14 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is commutative iff ❼c➁ a ❼b c➁ ➋ ✔ b ❼a c➁ ➋ (weak exchange condition).

Proposition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The followings are equivalent:

▲ ❼L,,➋➁ is commutative; ▲ the adjoint product ❛ is commutative; ▲ ❼

➋➁ ➝

• ❛ ❛

14 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

A w-ceo algebra ❼L,,➋➁ is commutative iff ❼c➁ a ❼b c➁ ➋ ✔ b ❼a c➁ ➋ (weak exchange condition).

Proposition

Let ❼L,,➋➁ be a right-distributive w-ceo algebra. The followings are equivalent:

▲ ❼L,,➋➁ is commutative; ▲ the adjoint product ❛ is commutative; ▲ ❼L,,➋➁ is symmetrical and ➝ coincides with (and, of

course, ˜ ❛ ❛).

14 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲

❼ ➋➁

▲ ▲

❛

▲ ❼ ➁

❼ ➁ ❼ ❛ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲ ❼

➝ ➋➁

▲

➝ ❼ ➝ ➁ ❼ ❛ ➁ ➝

▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲

❼ ➋➁

▲ ▲

❛

▲ ❼ ➁

❼ ➁ ❼ ❛ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲ ❼

➝ ➋➁

▲

➝ ❼ ➝ ➁ ❼ ❛ ➁ ➝

▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲

❼ ➋➁

▲ ❼

➋➁

▲ ❼

➝ ➋➁

▲

➝ ❼ ➝ ➁ ❼ ❛ ➁ ➝

▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼

➋➁

▲ ❼

➝ ➋➁

▲

➝ ❼ ➝ ➁ ❼ ❛ ➁ ➝

▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼

➝ ➋➁

▲

➝ ❼ ➝ ➁ ❼ ❛ ➁ ➝

▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼L, ➝, ➋➁ is associative. ▲

➝ ❼ ➝ ➁ ❼ ❛ ➁ ➝

▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼L, ➝, ➋➁ is associative. ▲ a ➝ ❼b ➝ c➁ ❼a ❛ b➁ ➝ c. (strong adjuction) ▲

➝ ❼ ➁ ❼ ➝ ➁

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼L, ➝, ➋➁ is associative. ▲ a ➝ ❼b ➝ c➁ ❼a ❛ b➁ ➝ c. (strong adjuction) ▲ a ➝ ❼b c➁ b ❼a ➝ c➁. (strong Galois connection)

▲

❼ ➋➁

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼L, ➝, ➋➁ is associative. ▲ a ➝ ❼b ➝ c➁ ❼a ❛ b➁ ➝ c. (strong adjuction) ▲ a ➝ ❼b c➁ b ❼a ➝ c➁. (strong Galois connection)

▲ If ❼L,,➋➁ is a symmetrical and commutative cdeo algebra,

then the following are equivalent:

▲ ❼

➋➁

▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼L, ➝, ➋➁ is associative. ▲ a ➝ ❼b ➝ c➁ ❼a ❛ b➁ ➝ c. (strong adjuction) ▲ a ➝ ❼b c➁ b ❼a ➝ c➁. (strong Galois connection)

▲ If ❼L,,➋➁ is a symmetrical and commutative cdeo algebra,

then the following are equivalent:

▲ ❼L, , ➋➁ is associative. ▲

❼ ➁ ❼ ➁

15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If ❼L,,➋➁ is a right-distributive w-ceo algebra, the following

are equivalent:

▲ L is associative; ▲ the adjoint product ❛ is associative; ▲ ❼a➁ a ❼b c➁ ❼b ❛ a➁ c. (strong adjunction)

▲ If ❼L,,➋➁ is a symmetrical cdeo algebra, the following are

equivalent:

▲ ❼L, , ➋➁ is associative. ▲ ❼L, ➝, ➋➁ is associative. ▲ a ➝ ❼b ➝ c➁ ❼a ❛ b➁ ➝ c. (strong adjuction) ▲ a ➝ ❼b c➁ b ❼a ➝ c➁. (strong Galois connection)

▲ If ❼L,,➋➁ is a symmetrical and commutative cdeo algebra,

then the following are equivalent:

▲ ❼L, , ➋➁ is associative. ▲ a ❼b c➁ b ❼a c➁. (strong exchange condition) 15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• B. J´
• nsson, A. Tarski: Representation problems for relation

algebras, Bull. Amer. Math. Soc. 54 (1948), 79-80. ❼ ✲ ✱ ✏ ❳ ✘ ➁ ❼ ➁

▲ ❼

✲ ✱ ✏ ➁

▲ ❼

❳ ➁

▲ ❼ ❳ ➁ ✱

• ✔ ❼ ✘ ❳ ➁ ✱
• ✔ ❼ ❳

✘➁ ✱

• ❼

➁

16 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• B. J´
• nsson, A. Tarski: Representation problems for relation

algebras, Bull. Amer. Math. Soc. 54 (1948), 79-80.

Definition

A (classical) relation algebra is a structure ❼A,✲,✱,✏ ,0,1,❳,✘ ,∆➁

• f type ❼2,2,1,0,0,2,1,0➁ such that:

▲ ❼A,✲,✱,✏ ,0,1➁ is a Boolean algebra; ▲ ❼A,❳,∆➁ is a monoid; ▲ ❼x ❳ y➁ ✱ z 0 ✔ ❼x✘ ❳ z➁ ✱ y 0 ✔ ❼z ❳ y ✘➁ ✱ x 0 (cycle

law). ❼ ➁

16 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• B. J´
• nsson, A. Tarski: Representation problems for relation

algebras, Bull. Amer. Math. Soc. 54 (1948), 79-80.

Definition

A (classical) relation algebra is a structure ❼A,✲,✱,✏ ,0,1,❳,✘ ,∆➁

• f type ❼2,2,1,0,0,2,1,0➁ such that:

▲ ❼A,✲,✱,✏ ,0,1➁ is a Boolean algebra; ▲ ❼A,❳,∆➁ is a monoid; ▲ ❼x ❳ y➁ ✱ z 0 ✔ ❼x✘ ❳ z➁ ✱ y 0 ✔ ❼z ❳ y ✘➁ ✱ x 0 (cycle

law).

Example

Rel❼X➁, the algebra of classical binary relations on a set X, is the standard example of relation algebra.

16 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• A. Popescu: Many-valued relation algebras, Algebra Universalis 53

(2005), 73-108. ❼ ❵ ❜ ✏ ❳ ✘ ➁ ❼ ➁

▲ ❼

❵ ❜ ✏ ➁

▲ ❼

❳ ➁

▲ ❼ ❳ ➁ ❜

• ✔ ❼ ✘ ❳ ➁ ❜
• ✔ ❼ ❳

✘➁ ❜

• ❼

➁ ❼ ✆

✕

❵ ❜ ✏ ❳ ✘ ➁

• ✆

▲ ▲ ❵ ❜ ✏

❼ ✆ ❵ ❜ ✏ ➁

▲ ❘ ❳ ❙❼

➁ ✝ ❃ ❘❼ ➁ ❜ ❙❼ ➁

▲ ❘✘❼

➁ ❘❼ ➁

▲

❼ ➁

• ❼

➁ ①

17 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• A. Popescu: Many-valued relation algebras, Algebra Universalis 53

(2005), 73-108.

Definition

An MV-relation algebra is a structure ❼A,❵,❜,✏ ,0,1,❳,✘ ,∆➁ of type ❼2,2,1,0,0,2,1,0➁ such that:

▲ ❼A,❵,❜,✏ ,0,1➁ is an MV-algebra; ▲ ❼A,❳,∆➁ is a monoid; ▲ ❼x ❳ y➁ ❜ z 0 ✔ ❼x✘ ❳ z➁ ❜ y 0 ✔ ❼z ❳ y ✘➁ ❜ x 0.

❼ ➁ ❼ ✆

✕

❵ ❜ ✏ ❳ ✘ ➁

• ✆

▲ ▲ ❵ ❜ ✏

❼ ✆ ❵ ❜ ✏ ➁

▲ ❘ ❳ ❙❼

➁ ✝ ❃ ❘❼ ➁ ❜ ❙❼ ➁

▲ ❘✘❼

➁ ❘❼ ➁

▲

❼ ➁

• ❼

➁ ①

17 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• A. Popescu: Many-valued relation algebras, Algebra Universalis 53

(2005), 73-108.

Definition

An MV-relation algebra is a structure ❼A,❵,❜,✏ ,0,1,❳,✘ ,∆➁ of type ❼2,2,1,0,0,2,1,0➁ such that:

▲ ❼A,❵,❜,✏ ,0,1➁ is an MV-algebra; ▲ ❼A,❳,∆➁ is a monoid; ▲ ❼x ❳ y➁ ❜ z 0 ✔ ❼x✘ ❳ z➁ ❜ y 0 ✔ ❼z ❳ y ✘➁ ❜ x 0.

Example

MVRel❼X➁ ❼0,1✆X✕X ,❵,❜,✏ ,0,1,❳,✘ ,∆➁ is the classical algebra of binary 0,1✆-relations on a set X where:

▲ 0 and 1 are the constant relations; ▲ ❵, ❜ and ✏ are the pointwise operation from ❼0,1✆,❵,❜,✏ ➁; ▲ ❘ ❳ ❙❼x,y➁ ✝z❃X ❘❼x,z➁ ❜ ❙❼z,y➁; ▲ ❘✘❼x,y➁ ❘❼y,x➁; ▲ ∆❼x,y➁ 1 if x y and ∆❼x,y➁ 0 if x ① y.

17 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• H. Furusawa, Y. Kawahara, M. Winter: Dedekind Categories with

Cutoff Operators, Fuzzy Sets Syst. (article in press). ❉

• ▲ ❉❼

➁ ❼❉❼ ➁ ❜ ✽ ✾ ✟ ➞ ➁ ➛ ❃ ❼❉➁

▲

❜

• ✾
• ✽

▲

✟ ❜ ✟ ✾ ❜

▲

➞

▲

✂ ❉❼ ➁ ❉❼ ➁ ➛

➐ ✂

❍ ➛ ✂ ❍

▲ ❼

• ➁
• ▲ ❼

➁

• ▲

❜

➐

❜

➐

18 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

• H. Furusawa, Y. Kawahara, M. Winter: Dedekind Categories with

Cutoff Operators, Fuzzy Sets Syst. (article in press).

Definition

A Dedekind category ❉ is a category with composition such that:

▲ ❉❼X,Y ➁ ❼❉❼X,Y ➁,❜,✽,✾,✟,0XY ,➞XY ➁ is an Heyting

algebra, ➛X,Y ❃ Obj❼❉➁,where:

▲ α ❜ β iff α α ✾ β iff β α ✽ β; ▲ α ✟ β is the relative pseudo-complement of α relative to β

i.e. γ ❜ α ✟ β iff α ✾ γ ❜ β;

▲ 0XY e ➞XY are the least and the greatest element.

▲ There exists a converse operation # ✂ ❉❼X,Y ➁ ❉❼Y ,X➁

such that ➛α,α➐ ✂ X ❍ Y , ➛β ✂ Y ❍ Z:

▲ ❼α β➁# β# α#; ▲ ❼α#➁# α; ▲ if α ❜ α➐, then α# ❜ α➐#. 18 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ ➛α ✂ X ❍ Y ,β ✂ Y ❍ Z,γ ✂ X ❍ Z:

α β ✾ γ ❜ α ❼β ✾ α# γ➁. ❼modular law➁

▲ ➛α ✂ X ❍ Y ,β ✂ Y ❍ Z the residual composition

α ❭ β ✂ X ❍ Z is a morphism such that ➛δ ✂ X ❍ Z, δ ❜ α ❭ β iff α# δ ❜ β. ❼ ➁❼ ➁ ❼ ❼ ➁❼ ➁ ❜ ✽ ✾ ✟ ➞ ➁ ❼ ✱ ✲ ➁

▲ ❜ ✽ ✾

✟ ✲ ✱ ❇

• ▲ ❘ ❼

➁ ❘❼ ➁

▲ ❘ ❙❼

➁ ✝ ❃ ❘❼ ➁ ✱ ❙❼ ➁

▲

• ➞
• ▲ ❘ ❭ ❙❼

➁ ☎ ❃ ❘❼ ➁ ❙❼ ➁

19 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

▲ ➛α ✂ X ❍ Y ,β ✂ Y ❍ Z,γ ✂ X ❍ Z:

α β ✾ γ ❜ α ❼β ✾ α# γ➁. ❼modular law➁

▲ ➛α ✂ X ❍ Y ,β ✂ Y ❍ Z the residual composition

α ❭ β ✂ X ❍ Z is a morphism such that ➛δ ✂ X ❍ Z, δ ❜ α ❭ β iff α# δ ❜ β.

Example

Rel❼L➁❼X,Y ➁ ❼Rel❼L➁❼X,Y ➁,❜,✽,✾,✟,0XY ,➞XY ➁ is the Dedekind category of binary heterogeneous L-relations taking value in an Heyting algebra ❼L,✱,✲,0,1,➁, where:

▲ ❜, ✽, ✾ and ✟ are pointwise induced by ✲, ✱, ❇ and of L; ▲ ❘#❼y,x➁ ❘❼x,y➁; ▲ ❘ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁ ✱ ❙❼y,z➁; ▲ 0XY 0 and ➞XY 1; ▲ ❘ ❭ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ❙❼y,z➁.

19 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Let L ❼L,,➋➁ be a w-ceo algebra and consider ➛X,Y ❃ ❙Set❙ the set R❼L➁❼X,Y ➁ of L-relations ❘ ✂ X ✕ Y L ✁ X ❍ Y .

What about R❼L➁❼X,Y ➁?

20 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ áXY ✂ X ❍ Y : áXY ❼x,y➁ ➊; ▲ ã

✂ ❍

ã

❼ ➁ ➋

▲ ■

✂ ❍ ■ ❼ ➁ ➋ ■ ❼

➐➁ ➊ ➛ ① ➐ ❃ ▲ ❘ ❇ ❘➐ ✔ ❘❼

➁ ❇ ❘➐❼ ➁

▲ ❘ ❘➐ ✂

❍ ❘ ❘➐❼ ➁ ❘❼ ➁ ❘➐❼ ➁ ❼ ➁❼ ➁ ❼ ❼ ➁❼ ➁ ã ➁

21 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ áXY ✂ X ❍ Y : áXY ❼x,y➁ ➊; ▲ ãXY ✂ X ❍ Y : ãXY ❼x,y➁ ➋; ▲ ■

✂ ❍ ■ ❼ ➁ ➋ ■ ❼

➐➁ ➊ ➛ ① ➐ ❃ ▲ ❘ ❇ ❘➐ ✔ ❘❼

➁ ❇ ❘➐❼ ➁

▲ ❘ ❘➐ ✂

❍ ❘ ❘➐❼ ➁ ❘❼ ➁ ❘➐❼ ➁ ❼ ➁❼ ➁ ❼ ❼ ➁❼ ➁ ã ➁

21 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ áXY ✂ X ❍ Y : áXY ❼x,y➁ ➊; ▲ ãXY ✂ X ❍ Y : ãXY ❼x,y➁ ➋; ▲ ■X ✂ X ❍ X : ■X❼x,x➁ ➋ and ■X❼x,x➐➁ ➊, ➛x ① x➐ ❃ X; ▲ ❘ ❇ ❘➐ ✔ ❘❼

➁ ❇ ❘➐❼ ➁

▲ ❘ ❘➐ ✂

❍ ❘ ❘➐❼ ➁ ❘❼ ➁ ❘➐❼ ➁ ❼ ➁❼ ➁ ❼ ❼ ➁❼ ➁ ã ➁

21 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ áXY ✂ X ❍ Y : áXY ❼x,y➁ ➊; ▲ ãXY ✂ X ❍ Y : ãXY ❼x,y➁ ➋; ▲ ■X ✂ X ❍ X : ■X❼x,x➁ ➋ and ■X❼x,x➐➁ ➊, ➛x ① x➐ ❃ X; ▲ ❘ ❇ ❘➐ ✔ ❘❼x,y➁ ❇ ❘➐❼x,y➁; ▲ ❘ ❘➐ ✂

❍ ❘ ❘➐❼ ➁ ❘❼ ➁ ❘➐❼ ➁ ❼ ➁❼ ➁ ❼ ❼ ➁❼ ➁ ã ➁

21 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ áXY ✂ X ❍ Y : áXY ❼x,y➁ ➊; ▲ ãXY ✂ X ❍ Y : ãXY ❼x,y➁ ➋; ▲ ■X ✂ X ❍ X : ■X❼x,x➁ ➋ and ■X❼x,x➐➁ ➊, ➛x ① x➐ ❃ X; ▲ ❘ ❇ ❘➐ ✔ ❘❼x,y➁ ❇ ❘➐❼x,y➁; ▲ ❘ ❘➐ ✂ X ❍ Y : ❘ ❘➐❼x,y➁ ❘❼x,y➁ ❘➐❼x,y➁.

❼ ➁❼ ➁ ❼ ❼ ➁❼ ➁ ã ➁

21 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

▲ áXY ✂ X ❍ Y : áXY ❼x,y➁ ➊; ▲ ãXY ✂ X ❍ Y : ãXY ❼x,y➁ ➋; ▲ ■X ✂ X ❍ X : ■X❼x,x➁ ➋ and ■X❼x,x➐➁ ➊, ➛x ① x➐ ❃ X; ▲ ❘ ❇ ❘➐ ✔ ❘❼x,y➁ ❇ ❘➐❼x,y➁; ▲ ❘ ❘➐ ✂ X ❍ Y : ❘ ❘➐❼x,y➁ ❘❼x,y➁ ❘➐❼x,y➁.

Consider R❼L➁❼X,Y ➁ ❼R❼L➁❼X,Y ➁,,ãXY ➁.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ If L is a w-ceo algebra, then R❼L➁❼X,Y ➁ is a w-ceo algebra; ▲ If L is a ceo algebra, then R❼L➁❼X,Y ➁ is a ceo algebra; ▲ If L is a right/left-distributive w-ceo algebra, then

R❼L➁❼X,Y ➁ is a right/left-distributive w-ceo algebra;

▲ If L is a cdeo algebra, then R❼L➁❼X,Y ➁ is a cdeo algebra; ▲ If L is symmetrical, then R❼L➁❼X,Y ➁ is symmetrical; ▲ If L is commutative, then R❼L➁❼X,Y ➁ is commutative; ▲ If L is associative, then R❼L➁❼X,Y ➁ is associative.

22 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036. ❼ ➋➁

▲ ❘✏ ✂

❍ ❘✏❼ ➁ ❘❼ ➁

▲ ❘ ❛ ❘➐ ✂

❍ ❘ ❛ ❘➐❼ ➁ ❘❼ ➁ ❛ ❘➐❼ ➁

▲ ❘☎ ❙ ✂

❍ ❘☎ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ❙❼ ➁

▲ ❘☎✂ ❙ ✂

❍ ❘☎✂ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ❘❼ ➁

▲ ❘✝❛ ❙ ✂

❍ ❼❘✝❛ ❙➁❼ ➁ ✝ ❃ ❘❼ ➁ ❛ ❙❼ ➁ ❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

23 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂

❍ ❘✏❼ ➁ ❘❼ ➁

▲ ❘ ❛ ❘➐ ✂

❍ ❘ ❛ ❘➐❼ ➁ ❘❼ ➁ ❛ ❘➐❼ ➁

▲ ❘☎ ❙ ✂

❍ ❘☎ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ❙❼ ➁

▲ ❘☎✂ ❙ ✂

❍ ❘☎✂ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ❘❼ ➁

▲ ❘✝❛ ❙ ✂

❍ ❼❘✝❛ ❙➁❼ ➁ ✝ ❃ ❘❼ ➁ ❛ ❙❼ ➁ ❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

23 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂ Y ❍ X: ❘✏❼y,x➁ ❘❼x,y➁; ▲ ❘ ❛ ❘➐ ✂

❍ ❘ ❛ ❘➐❼ ➁ ❘❼ ➁ ❛ ❘➐❼ ➁

▲ ❘☎ ❙ ✂

❍ ❘☎ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ❙❼ ➁

▲ ❘☎✂ ❙ ✂

❍ ❘☎✂ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ❘❼ ➁

▲ ❘✝❛ ❙ ✂

❍ ❼❘✝❛ ❙➁❼ ➁ ✝ ❃ ❘❼ ➁ ❛ ❙❼ ➁ ❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

23 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂ Y ❍ X: ❘✏❼y,x➁ ❘❼x,y➁; ▲ ❘ ❛ ❘➐ ✂ X ❍ Y : ❘ ❛ ❘➐❼x,y➁ ❘❼x,y➁ ❛ ❘➐❼x,y➁; ▲ ❘☎ ❙ ✂

❍ ❘☎ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ❙❼ ➁

▲ ❘☎✂ ❙ ✂

❍ ❘☎✂ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ❘❼ ➁

▲ ❘✝❛ ❙ ✂

❍ ❼❘✝❛ ❙➁❼ ➁ ✝ ❃ ❘❼ ➁ ❛ ❙❼ ➁ ❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

23 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂ Y ❍ X: ❘✏❼y,x➁ ❘❼x,y➁; ▲ ❘ ❛ ❘➐ ✂ X ❍ Y : ❘ ❛ ❘➐❼x,y➁ ❘❼x,y➁ ❛ ❘➐❼x,y➁; ▲ ❘☎ ❙ ✂ X ❍ Z: ❘☎ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ❙❼y,z➁; ▲ ❘☎✂ ❙ ✂

❍ ❘☎✂ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ❘❼ ➁

▲ ❘✝❛ ❙ ✂

❍ ❼❘✝❛ ❙➁❼ ➁ ✝ ❃ ❘❼ ➁ ❛ ❙❼ ➁ ❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

23 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂ Y ❍ X: ❘✏❼y,x➁ ❘❼x,y➁; ▲ ❘ ❛ ❘➐ ✂ X ❍ Y : ❘ ❛ ❘➐❼x,y➁ ❘❼x,y➁ ❛ ❘➐❼x,y➁; ▲ ❘☎ ❙ ✂ X ❍ Z: ❘☎ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ❙❼y,z➁; ▲ ❘☎✂ ❙ ✂ X ❍ Z: ❘☎✂ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ❘❼x,y➁; ▲ ❘✝❛ ❙ ✂

❍ ❼❘✝❛ ❙➁❼ ➁ ✝ ❃ ❘❼ ➁ ❛ ❙❼ ➁ ❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

23 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂ Y ❍ X: ❘✏❼y,x➁ ❘❼x,y➁; ▲ ❘ ❛ ❘➐ ✂ X ❍ Y : ❘ ❛ ❘➐❼x,y➁ ❘❼x,y➁ ❛ ❘➐❼x,y➁; ▲ ❘☎ ❙ ✂ X ❍ Z: ❘☎ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ❙❼y,z➁; ▲ ❘☎✂ ❙ ✂ X ❍ Z: ❘☎✂ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ❘❼x,y➁; ▲ ❘✝❛ ❙ ✂ X ❍ Z: ❼❘✝❛ ❙➁❼x,z➁ ✝y❃Y ❘❼x,y➁ ❛ ❙❼y,z➁.

❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

R.Bˇ elohl´ avek: Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, Plenum Press, Dordrecht, NewYork, 2002.

• L. Bˇ

ehounek, M. Daˇ nkov´ a: Relational compositions in Fuzzy Class Theory, Fuzzy Sets Syst. 160 (2008), 1005-1036.

Definition

Let ❼L,,➋➁ be a cdeo algebra.

▲ ❘✏ ✂ Y ❍ X: ❘✏❼y,x➁ ❘❼x,y➁; ▲ ❘ ❛ ❘➐ ✂ X ❍ Y : ❘ ❛ ❘➐❼x,y➁ ❘❼x,y➁ ❛ ❘➐❼x,y➁; ▲ ❘☎ ❙ ✂ X ❍ Z: ❘☎ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ❙❼y,z➁; ▲ ❘☎✂ ❙ ✂ X ❍ Z: ❘☎✂ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ❘❼x,y➁; ▲ ❘✝❛ ❙ ✂ X ❍ Z: ❼❘✝❛ ❙➁❼x,z➁ ✝y❃Y ❘❼x,y➁ ❛ ❙❼y,z➁.

Remark

❘☎✂ ❙ ❼❙✏ ☎ ❘✏➁✏.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘ ▲

❘ ❇ ❘➐ ❘✏ ❇ ❘➐

✏ ▲ ❘☎ ã

ã

▲ á

☎ ❙ ã

▲ ❘❛ ã

❘

▲ ❘✝❛ ■

❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲

❘ ❇ ❘➐ ❘✏ ❇ ❘➐

✏ ▲ ❘☎ ã

ã

▲ á

☎ ❙ ã

▲ ❘❛ ã

❘

▲ ❘✝❛ ■

❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ã

ã

▲ á

☎ ❙ ã

▲ ❘❛ ã

❘

▲ ❘✝❛ ■

❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ á

☎ ❙ ã

▲ ❘❛ ã

❘

▲ ❘✝❛ ■

❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ã

❘

▲ ❘✝❛ ■

❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ãXY ❘; ▲ ❘✝❛ ■

❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ãXY ❘; ▲ ❘✝❛ ■Y ❘; ▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

24 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ãXY ❘; ▲ ❘✝❛ ■Y ❘; ▲ ❘✝❛áYZ áXY ; ▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ãXY ❘; ▲ ❘✝❛ ■Y ❘; ▲ ❘✝❛áYZ áXY ; ▲ áXY ✝❛ ❙ áXZ; ▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ãXY ❘; ▲ ❘✝❛ ■Y ❘; ▲ ❘✝❛áYZ áXY ; ▲ áXY ✝❛ ❙ áXZ; ▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙;(right-residual composition) ▲

❼❘✝❛ ❙➁ ❛ ❚ ã ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ã

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ❼❘✏➁✏ ❘; ▲ if ❘ ❇ ❘➐, then ❘✏ ❇ ❘➐ ✏; ▲ ❘☎ ãYZ ãXZ; ▲ áXY ☎ ❙ ãXZ; ▲ ❘❛ ãXY ❘; ▲ ❘✝❛ ■Y ❘; ▲ ❘✝❛áYZ áXY ; ▲ áXY ✝❛ ❙ áXZ; ▲ ❚ ❇ ❘☎ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙;(right-residual composition) ▲ if ❼❘✝❛ ❙➁ ❛ ❚ ãXZ, then ❘✝❛❼❙ ❛ ❘✏ ✝❛ ❚ ➁ ãXZ;

(weak modular law)

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂

❍ ❘ ➝ ❘➐❼ ➁ ❘❼ ➁ ➝ ❘➐❼ ➁

▲ ❘❛❘➐ ✂

❍ ❘❛❘➐❼ ➁ ❘❼ ➁❛❘➐❼ ➁

▲ ❘✝❛

✂ ❍ ❘✝➬

❛ ❙❼

➁ ✝ ❃ ❘❼ ➁➬ ❛❙❼ ➁

▲ ❘☎➝ ❙ ✂

❍ ❘☎➝ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ➝ ❙❼ ➁

▲ ❘☎➟ ❙ ✂

❍ ❘☎➟ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ➟ ❘❼ ➁

▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘❛❘➐ ✂

❍ ❘❛❘➐❼ ➁ ❘❼ ➁❛❘➐❼ ➁

▲ ❘✝❛

✂ ❍ ❘✝➬

❛ ❙❼

➁ ✝ ❃ ❘❼ ➁➬ ❛❙❼ ➁

▲ ❘☎➝ ❙ ✂

❍ ❘☎➝ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ➝ ❙❼ ➁

▲ ❘☎➟ ❙ ✂

❍ ❘☎➟ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ➟ ❘❼ ➁

▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝❛

✂ ❍ ❘✝➬

❛ ❙❼

➁ ✝ ❃ ❘❼ ➁➬ ❛❙❼ ➁

▲ ❘☎➝ ❙ ✂

❍ ❘☎➝ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ➝ ❙❼ ➁

▲ ❘☎➟ ❙ ✂

❍ ❘☎➟ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ➟ ❘❼ ➁

▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝˜ ❛ S ✂ X ❍ Z: ❘✝➬ ❛ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁➬

❛❙❼y,z➁;

▲ ❘☎➝ ❙ ✂

❍ ❘☎➝ ❙❼ ➁ ☎ ❃ ❘❼ ➁ ➝ ❙❼ ➁

▲ ❘☎➟ ❙ ✂

❍ ❘☎➟ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ➟ ❘❼ ➁

▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝˜ ❛ S ✂ X ❍ Z: ❘✝➬ ❛ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁➬

❛❙❼y,z➁;

▲ ❘☎➝ ❙ ✂ X ❍ Z: ❘☎➝ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ➝ ❙❼y,z➁; ▲ ❘☎➟ ❙ ✂

❍ ❘☎➟ ❙❼ ➁ ☎ ❃ ❙❼ ➁ ➟ ❘❼ ➁

▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝˜ ❛ S ✂ X ❍ Z: ❘✝➬ ❛ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁➬

❛❙❼y,z➁;

▲ ❘☎➝ ❙ ✂ X ❍ Z: ❘☎➝ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ➝ ❙❼y,z➁; ▲ ❘☎➟ ❙ ✂ X ❍ Z: ❘☎➟ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ➟ ❘❼x,y➁; ▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝˜ ❛ S ✂ X ❍ Z: ❘✝➬ ❛ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁➬

❛❙❼y,z➁;

▲ ❘☎➝ ❙ ✂ X ❍ Z: ❘☎➝ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ➝ ❙❼y,z➁; ▲ ❘☎➟ ❙ ✂ X ❍ Z: ❘☎➟ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ➟ ❘❼x,y➁;

Remark

▲ ❼❘✝❛ ❙➁✏ ❙✏ ✝❛ ❘✏ ▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝˜ ❛ S ✂ X ❍ Z: ❘✝➬ ❛ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁➬

❛❙❼y,z➁;

▲ ❘☎➝ ❙ ✂ X ❍ Z: ❘☎➝ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ➝ ❙❼y,z➁; ▲ ❘☎➟ ❙ ✂ X ❍ Z: ❘☎➟ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ➟ ❘❼x,y➁;

Remark

▲ ❼❘✝˜ ❛ ❙➁✏ ❙✏ ✝❛ ❘✏; (extend a condition of Dedekind

category)

▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼L,,➋➁ be a symmetrical cdeo algebra.

▲ ❘ ➝ ❘➐ ✂ X ❍ Y : ❘ ➝ ❘➐❼x,y➁ ❘❼x,y➁ ➝ ❘➐❼x,y➁; ▲ ❘˜

❛❘➐ ✂ X ❍ Y : ❘˜ ❛❘➐❼x,y➁ ❘❼x,y➁˜ ❛❘➐❼x,y➁;

▲ ❘✝˜ ❛ S ✂ X ❍ Z: ❘✝➬ ❛ ❙❼x,z➁ ✝y❃Y ❘❼x,y➁➬

❛❙❼y,z➁;

▲ ❘☎➝ ❙ ✂ X ❍ Z: ❘☎➝ ❙❼x,z➁ ☎y❃Y ❘❼x,y➁ ➝ ❙❼y,z➁; ▲ ❘☎➟ ❙ ✂ X ❍ Z: ❘☎➟ ❙❼x,z➁ ☎y❃Y ❙❼y,z➁ ➟ ❘❼x,y➁;

Remark

▲ ❼❘✝˜ ❛ ❙➁✏ ❙✏ ✝❛ ❘✏; (extend a condition of Dedekind

category)

▲ ❘☎➟ ❙ ❼❙✏ ☎➝ ❘✏➁✏.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■ ✝❛ ❘ ❘ ▲ ã

❛❘ ❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ã

❛❘ ❘

▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ãXY ❛❘ ❘; ▲ ❘✝❛á

á

▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ãXY ❛❘ ❘; ▲ ❘✝˜ ❛áYZ áXY ; ▲ á

✝❛ ❙ á

▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ãXY ❛❘ ❘; ▲ ❘✝˜ ❛áYZ áXY ; ▲ áXY ✝˜ ❛ ❙ áXZ; ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝❛ ❚ ❇ ❙ ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ãXY ❛❘ ❘; ▲ ❘✝˜ ❛áYZ áXY ; ▲ áXY ✝˜ ❛ ❙ áXZ; ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝˜ ❛ ❚ ❇ ❙ (left-residual composition); ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ãXY ❛❘ ❘; ▲ ❘✝˜ ❛áYZ áXY ; ▲ áXY ✝˜ ❛ ❙ áXZ; ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝˜ ❛ ❚ ❇ ❙ (left-residual composition); ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ (Galois connection); ▲

❼❘✝❛ ❙➁❛❚ ã ❘✝❛❼❙❛❘✏ ✝❛ ❚ ➁ ã ❛

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

▲ ■X ✝❛ ❘ ❘; ▲ ãXY ❛❘ ❘; ▲ ❘✝˜ ❛áYZ áXY ; ▲ áXY ✝˜ ❛ ❙ áXZ; ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘✏ ✝˜ ❛ ❚ ❇ ❙ (left-residual composition); ▲ ❚ ❇ ❘☎➝ ❙ ✔ ❘ ❇ ❚ ☎ ❙✏ (Galois connection); ▲ if ❼❘✝˜ ❛ ❙➁˜

❛❚ ãXZ, then ❘✝˜

❛❼❙ ˜

❛❘✏ ✝˜

❛ ❚ ➁ ãXZ;

(weak modular law for ˜ ❛)

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

If moreover L is associative, then the followings hold:

▲ ❘✝❛❼❙ ✝❛ ◗➁ ❼❘✝❛ ❙➁✝❛ ◗; ▲ ❘✝❛❼❙ ✝❛ ◗➁ ❼❘✝❛ ❙➁✝❛ ◗ ▲ ❘☎❼❙ ☎ ❚ ➁ ❼❘✝❛ ❙➁☎ ❚ ▲ ❘☎➝❼❙ ☎➝ ❚ ➁ ❼❘✝❛ ❙➁☎➝ ❚ ▲ ❼❘✝❛ ❙➁ ❛ ❚ á

✔ ❼❚ ✝ ➬ ❛❙✏➁➬ ❛❘ á

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If moreover L is associative, then the followings hold:

▲ ❘✝❛❼❙ ✝❛ ◗➁ ❼❘✝❛ ❙➁✝❛ ◗; ▲ ❘✝˜ ❛❼❙ ✝˜ ❛ ◗➁ ❼❘✝˜ ❛ ❙➁✝˜ ❛ ◗; ▲ ❘☎❼❙ ☎ ❚ ➁ ❼❘✝❛ ❙➁☎ ❚ ▲ ❘☎➝❼❙ ☎➝ ❚ ➁ ❼❘✝❛ ❙➁☎➝ ❚ ▲ ❼❘✝❛ ❙➁ ❛ ❚ á

✔ ❼❚ ✝ ➬ ❛❙✏➁➬ ❛❘ á

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If moreover L is associative, then the followings hold:

▲ ❘✝❛❼❙ ✝❛ ◗➁ ❼❘✝❛ ❙➁✝❛ ◗; ▲ ❘✝˜ ❛❼❙ ✝˜ ❛ ◗➁ ❼❘✝˜ ❛ ❙➁✝˜ ❛ ◗; ▲ ❘☎❼❙ ☎ ❚ ➁ ❼❘✝˜ ❛ ❙➁☎ ❚ ; ▲ ❘☎➝❼❙ ☎➝ ❚ ➁ ❼❘✝❛ ❙➁☎➝ ❚ ▲ ❼❘✝❛ ❙➁ ❛ ❚ á

✔ ❼❚ ✝ ➬ ❛❙✏➁➬ ❛❘ á

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

If moreover L is associative, then the followings hold:

▲ ❘✝❛❼❙ ✝❛ ◗➁ ❼❘✝❛ ❙➁✝❛ ◗; ▲ ❘✝˜ ❛❼❙ ✝˜ ❛ ◗➁ ❼❘✝˜ ❛ ❙➁✝˜ ❛ ◗; ▲ ❘☎❼❙ ☎ ❚ ➁ ❼❘✝˜ ❛ ❙➁☎ ❚ ; ▲ ❘☎➝❼❙ ☎➝ ❚ ➁ ❼❘✝❛ ❙➁☎➝ ❚ ; ▲ ❼❘✝❛ ❙➁ ❛ ❚ á

✔ ❼❚ ✝ ➬ ❛❙✏➁➬ ❛❘ á

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

If moreover L is associative, then the followings hold:

▲ ❘✝❛❼❙ ✝❛ ◗➁ ❼❘✝❛ ❙➁✝❛ ◗; ▲ ❘✝˜ ❛❼❙ ✝˜ ❛ ◗➁ ❼❘✝˜ ❛ ❙➁✝˜ ❛ ◗; ▲ ❘☎❼❙ ☎ ❚ ➁ ❼❘✝˜ ❛ ❙➁☎ ❚ ; ▲ ❘☎➝❼❙ ☎➝ ❚ ➁ ❼❘✝❛ ❙➁☎➝ ❚ ; ▲ ❼❘✝❛ ❙➁ ❛ ❚ áXZ✔ ❼❚ ✝ ➬

❛❙✏➁➬ ❛❘ áXY .(one equivalence of cycle law)

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

We need to extend the notion of category and hence we propose the following definition.

Definition

A pseudo bi-category ❈ ❼❖❼❈➁, ▼❼❈➁,❳,˜ ❳,■,˜ ■➁ consists in a class of objects, a class of morphisms, two partial composition

• perations and two families of identities (in the usual sense) where

we require only:

▲ right neutrality of the identities ■X with respect to ❳; ▲ left neutrality of the identities ˜

■X with respect to ˜ ❳; Note that the associativity of the compositions is not required. ❈

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We need to extend the notion of category and hence we propose the following definition.

Definition

A pseudo bi-category ❈ ❼❖❼❈➁, ▼❼❈➁,❳,˜ ❳,■,˜ ■➁ consists in a class of objects, a class of morphisms, two partial composition

• perations and two families of identities (in the usual sense) where

we require only:

▲ right neutrality of the identities ■X with respect to ❳; ▲ left neutrality of the identities ˜

■X with respect to ˜ ❳; Note that the associativity of the compositions is not required.

Definition

A pseudo bi-category ❈ is called bi-category if the two partial compositions are associative.

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Definition

We say that a pseudo bi-category is symmetrical if ➛X ❃❖❼❈➁ ■X ˜ ■X and there exists the opposite operation ✆✏ ✂▼❼❈➁▼❼❈➁, α ❃ ❈❼X,Y ➁ ✭ α✏ ❃ ❈❼Y ,X➁ such that ➛α,β ❃▼❼❈➁: ❼α˜ ❳β➁✏ β✏ ❳ α✏.

❼ ➁

❼ ➋➁ ❼ ➁ ✝❛ ✝❛ ❼ ➋➁ ❼ ➁

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Definition

We say that a pseudo bi-category is symmetrical if ➛X ❃❖❼❈➁ ■X ˜ ■X and there exists the opposite operation ✆✏ ✂▼❼❈➁▼❼❈➁, α ❃ ❈❼X,Y ➁ ✭ α✏ ❃ ❈❼Y ,X➁ such that ➛α,β ❃▼❼❈➁: ❼α˜ ❳β➁✏ β✏ ❳ α✏.

What about R❼L➁?

❼ ➋➁ ❼ ➁ ✝❛ ✝❛ ❼ ➋➁ ❼ ➁

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

We say that a pseudo bi-category is symmetrical if ➛X ❃❖❼❈➁ ■X ˜ ■X and there exists the opposite operation ✆✏ ✂▼❼❈➁▼❼❈➁, α ❃ ❈❼X,Y ➁ ✭ α✏ ❃ ❈❼Y ,X➁ such that ➛α,β ❃▼❼❈➁: ❼α˜ ❳β➁✏ β✏ ❳ α✏.

What about R❼L➁?

Hence, if ❼L,,➋➁ is a symmetrical cdeo algebra, then the class R❼L➁ of all L-binary relations between sets with the two compositions ✝❛ and ✝˜

❛ is a symmetrical pseudo bi-category that

is a weak and non-commutative generalization of a Dedekind

• category. Moreover, if ❼L,,➋➁ is a symmetrical and associative

cdeo algebra, R❼L➁ becomes, of course,a bi-category.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Looking at ❇ in a symmetrical cdeo algebra as a (crisp) L-relation and at the operations , ❛ and ➝ as L-relations, one has: ❼L,❇➁

❛

• ❼L,❇➁

➝

• ❼L,❇➁
• ❇ ❼a,c ➝ b➁ ➋ ✔ ❇ ❼a ❛ c,b➁ ➋ ✔ ❇ ❼c,a b➁ ➋

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼A,❘A➁, ❼B,❘B➁ and ❼C,❘C➁ be sets equipped with fixed L-relations, where L ❼L,,➋➁ is a symmetrical cdeo algebra. The diagram ❼C,❘C➁

χ

• ❼A,❘A➁

ψ

• ❼B,❘B➁

ϕ

• with ϕ ❃ R❼C➁❼A,B➁, i.e. ϕ is a C-valued binary relation from A

to B, ψ ❃ R❼B➁❼A,C➁ and χ ❃ R❼A➁❼B,C➁ is a weak L-Galois triangle if for all a ❃ A,b ❃ B,c ❃ C the following equivalences hold: ❘A❼a,χ❼b,c➁➁ ➋ ✔ ❘B❼ψ❼a,c➁,b➁ ➋ ✔ ❘C❼c,ϕ❼a,b➁➁ ➋.

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Proposition

Let L ❼L,,➋➁ be a symmetrical cdeo algebra. Then for any triple of sets ❼X,Y ,Z➁ one has a weak L-Galois triangle ❼R❼L➁❼X,Y ➁,❇➁

χ

• ❼R❼L➁❼Z,X➁,❇➁

ψ

• ❼R❼L➁❼Y ,Z➁,❇➁

ϕ

• where

ϕ ❃ R❼R❼L➁❼X,Y ➁➁❼R❼Z,X➁,R❼Y ,Z➁➁, ψ ❃ R❼R❼L➁❼Y ,Z➁➁❼R❼Z,X➁,R❼X,Y ➁➁, χ ❃ R❼R❼L➁❼Z,X➁➁❼R❼Y ,Z➁,R❼X,Y ➁➁ are defined, ➛α ❃ R❼L➁❼Z,X➁,β ❃ R❼L➁❼Y ,Z➁,γ ❃ R❼L➁❼X,Y ➁ by: ❨ ϕ❼α,β➁ ❼β ☎✂ α➁✏; ❨ ψ❼α,γ➁ ❼α ✝❛ γ➁✏; ❨ χ❼β,γ➁ ❼γ ☎➝ β➁✏.

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If the symmetrical cdeo algebra ❼L,,➋➁ is associative, one has stronger conditions on the triangle ❼L,,➝➁

✂ ➟

• ❼L,,➝➁

˜ ❛ ❛

• ❼L,,➝➁

➝

• ▲ a ❼b ✂ c➁ ❼a˜

❛c➁ b;

▲ a ➝ ❼b ➟ c➁ ❼a ❛ c➁ ➝ b; ▲ a ❼b ➟ c➁ c ➝ ❼a b➁; ▲ a ➝ ❼b ✂ b➁ c ❼a ➝ b➁.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼A,❘A, ˜ ❘A➁, ❼B,❘B, ˜ ❘B➁ and ❼C,❘C, ˜ ❘C➁ be the sets equipped with two fixed L-relations, where L ❼L,,➋➁ is a symmetrical cdeo algebra. The diagram ❼C,❘C, ˜ ❘C➁

χ ˜ χ

• ❼A,❘A, ˜

❘A➁

ψ ˜ ψ

• ❼B,❘B, ˜

❘B➁

ϕ ˜ ϕ

• with

ϕ, ˜ ϕ ❃ R❼C➁❼A,B➁, ψ, ˜ ψ ❃ R❼B➁❼A,C➁ and χ, ˜ χ ❃ R❼A➁❼B,C➁ is a symmetrical L-Galois triangle if for all a ❃ A,b ❃ B,c ❃ C the following equalities hold:

• 1. ❘A❼a,χ❼b,c➁➁ ❘B❼ ˜

ψ❼a,c➁,b➁;

• 2. ˜

❘A❼a, ˜ χ❼b,c➁➁ ˜ ❘B❼ψ❼a,c➁,b➁;

• 3. ❘A❼a, ˜

χ❼b,c➁➁ ˜ ❘C❼c,ϕ❼a,b➁➁;

• 4. ˜

❘A❼a,χ❼b,c➁➁ ❘C❼c, ˜ ϕ❼a,b➁➁.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. ❙ ✂ ✕

• ❙❼

➁ ☎ ❃ ❼ ➁ ❼ ➁ ❃ ❚ ✂ ✕

• ❚ ❼

➁ ✝ ❃ ❼ ➁ ❛ ❼ ➁ ❃ ❼ ➋➁ ❙ ❼ ➁ ☎ ❃ ❼ ➁ ➝ ❼ ➁ ❃ ❚ ❼ ➁ ✝ ❃ ❼ ➁❛ ❼ ➁ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. The subsethood degree is the L-relation ❙X ✂ LX ✕ LX L defined by: ❙❼ ➁ ☎ ❃ ❼ ➁ ❼ ➁ ❃ ❚ ✂ ✕

• ❚ ❼

➁ ✝ ❃ ❼ ➁ ❛ ❼ ➁ ❃ ❼ ➋➁ ❙ ❼ ➁ ☎ ❃ ❼ ➁ ➝ ❼ ➁ ❃ ❚ ❼ ➁ ✝ ❃ ❼ ➁❛ ❼ ➁ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. The subsethood degree is the L-relation ❙X ✂ LX ✕ LX L defined by: ❙❼A,B➁ ☎x❃X A❼x➁ B❼x➁, for all A,B ❃ LX. ❚ ✂ ✕

• ❚ ❼

➁ ✝ ❃ ❼ ➁ ❛ ❼ ➁ ❃ ❼ ➋➁ ❙ ❼ ➁ ☎ ❃ ❼ ➁ ➝ ❼ ➁ ❃ ❚ ❼ ➁ ✝ ❃ ❼ ➁❛ ❼ ➁ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. The subsethood degree is the L-relation ❙X ✂ LX ✕ LX L defined by: ❙❼A,B➁ ☎x❃X A❼x➁ B❼x➁, for all A,B ❃ LX. The overlap degree is the L-relation ❚X ✂ LX ✕ LX L defined by: ❚ ❼ ➁ ✝ ❃ ❼ ➁ ❛ ❼ ➁ ❃ ❼ ➋➁ ❙ ❼ ➁ ☎ ❃ ❼ ➁ ➝ ❼ ➁ ❃ ❚ ❼ ➁ ✝ ❃ ❼ ➁❛ ❼ ➁ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. The subsethood degree is the L-relation ❙X ✂ LX ✕ LX L defined by: ❙❼A,B➁ ☎x❃X A❼x➁ B❼x➁, for all A,B ❃ LX. The overlap degree is the L-relation ❚X ✂ LX ✕ LX L defined by: ❚X❼A,B➁ ✝x❃X A❼x➁ ❛ B❼x➁, for all A,B ❃ LX. ❼ ➋➁ ❙ ❼ ➁ ☎ ❃ ❼ ➁ ➝ ❼ ➁ ❃ ❚ ❼ ➁ ✝ ❃ ❼ ➁❛ ❼ ➁ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. The subsethood degree is the L-relation ❙X ✂ LX ✕ LX L defined by: ❙❼A,B➁ ☎x❃X A❼x➁ B❼x➁, for all A,B ❃ LX. The overlap degree is the L-relation ❚X ✂ LX ✕ LX L defined by: ❚X❼A,B➁ ✝x❃X A❼x➁ ❛ B❼x➁, for all A,B ❃ LX. If ❼L,,➋➁ is a symmetrical cdeo algebra, we consider further L-relations: ❙ ❼ ➁ ☎ ❃ ❼ ➁ ➝ ❼ ➁ ❃ ❚ ❼ ➁ ✝ ❃ ❼ ➁❛ ❼ ➁ ❃

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let X be a set and let ❼L,,➋➁ be a w-ceo algebra. The subsethood degree is the L-relation ❙X ✂ LX ✕ LX L defined by: ❙❼A,B➁ ☎x❃X A❼x➁ B❼x➁, for all A,B ❃ LX. The overlap degree is the L-relation ❚X ✂ LX ✕ LX L defined by: ❚X❼A,B➁ ✝x❃X A❼x➁ ❛ B❼x➁, for all A,B ❃ LX. If ❼L,,➋➁ is a symmetrical cdeo algebra, we consider further L-relations: ˜ ❙X❼A,B➁ ☎x❃X A❼x➁ ➝ B❼x➁, for all A,B ❃ LX ˜ ❚X❼A,B➁ ✝x❃X A❼x➁˜ ❛B❼x➁, for all A,B ❃ LX.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

Let L ❼L,,➋➁ be a symmetrical cdeo algebra. Then for any triple of sets ❼X,Y ,Z➁ the diagram ❼R❼L➁❼X,Y ➁,❙XY , ˜ ❙XY ➁

χ ˜ χ

• ❼R❼L➁❼Z,X➁,❙ZX, ˜

❙ZX➁

ψ ˜ ψ

• ❼R❼L➁❼Y ,Z➁,❙YZ, ˜

❙YZ➁

ϕ ˜ ϕ

• where

ϕ, ˜ ϕ ❃ R❼R❼L➁❼X,Y ➁➁❼R❼L➁❼Z,X➁,R❼L➁❼Y ,Z➁➁, ψ, ˜ ψ ❃ R❼R❼L➁❼Y ,Z➁➁❼R❼L➁❼Z,X➁,R❼L➁❼X,Y ➁➁, χ, ˜ χ ❃ R❼R❼L➁❼Z,X➁➁❼R❼L➁❼Y ,Z➁,R❼L➁❼X,Y ➁➁ are defined ➛α ❃ R❼L➁❼Z,X➁,β ❃ R❼L➁❼Y ,Z➁,γ ❃ R❼L➁❼X,Y ➁ by: ❨ ϕ❼α,β➁ α✏ ☎ β✏; ˜ ϕ❼α,β➁ α✏ ☎➝ β✏; ❨ψ❼α,γ➁ ❼α ✝❛ γ➁✏ ˜ ψ❼α,γ➁ ❼α ✝˜

❛ γ➁✏;

❨ χ❼β,γ➁ β✏ ☎✂ γ✏; ˜ χ❼β,γ➁ β✏ ☎➟ γ✏; is a symmetrical L-Galois triangle if and only if L is associative.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

Let L ❼L,,➋➁ be a symmetrical cdeo algebra. Then for any triple of sets ❼X,Y ,Z➁ the diagram ❼R❼L➁❼X,Y ➁,❚XY , ˜ ❚XY ➁

χ ˜ χ

• ❼R❼L➁❼Z,X➁,❚ZX, ˜

❚ZX➁

ψ ˜ ψ

• ❼R❼L➁❼Y ,Z➁,❚YZ, ˜

❚YZ➁

ϕ ˜ ϕ

• where

ϕ, ˜ ϕ ❃ R❼R❼L➁❼X,Y ➁➁❼R❼L➁❼Z,X➁,R❼L➁❼Y ,Z➁➁, ψ, ˜ ψ ❃ R❼R❼L➁❼Y ,Z➁➁❼R❼L➁❼Z,X➁,R❼L➁❼X,Y ➁➁, χ, ˜ χ ❃ R❼R❼L➁❼Z,X➁➁❼R❼L➁❼Y ,Z➁,R❼L➁❼X,Y ➁➁ are defined, ➛α ❃ R❼L➁❼Z,X➁,β ❃ R❼L➁❼Y ,Z➁,γ ❃ R❼L➁❼X,Y ➁ by: ❨ ϕ❼α,β➁ α✏ ✝❛ β✏; ˜ ϕ❼α,β➁ α✏ ✝˜

❛ β;

❨ ψ❼α,γ➁ ❼α ✝❛ γ➁✏; ˜ ψ❼α,γ➁ ❼α ✝˜

❛ γ➁✏;

❨χ❼β,γ➁ ❼γ ✝❛ β➁✏; ˜ χ❼β,γ➁ ❼γ ✝˜

❛ β➁✏,

is a symmetrical L-Galois triangle if and only if L is associative.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

If the symmetrical cdeo algebra ❼L,,➋➁ is associative and commutative, one has stronger conditions on the triangle ❼L,➁

✂

• ❼L,➁

❛

• ❼L,➁
• a ❼c b➁ ❼c ❛ a➁ b c ❼b ✂ a➁.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Definition

Let ❼A,❘A➁, ❼B,❘B➁ and ❼C,❘C➁ be sets equipped with a fixed L-relation, where L ❼L,,➋➁ is a symmetrical cdeo algebra. The diagram ❼C,❘C➁

χ

• ❼A,❘A➁

ψ

• ❼B,❘B➁

ϕ

• with ϕ ❃ R❼C➁❼A,B➁, ψ ❃ R❼B➁❼A,C➁ and χ ❃ R❼A➁❼B,C➁ is a

strong L-Galois triangle if for all a ❃ A,b ❃ B,c ❃ C the following equalities hold: ❘A❼a,χ❼b,c➁➁ ❘B❼ψ❼a,c➁,b➁ ❘C❼c,ϕ❼a,b➁➁.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

Let L ❼L,,➋➁ be a symmetrical cdeo algebra. Then for any triple of sets ❼X,Y ,Z➁ the diagram ❼R❼L➁❼X,Y ➁,❙XY ➁

χ

• ❼R❼L➁❼Z,X➁,❙ZX➁

ψ

• ❼R❼L➁❼Y ,Z➁,❙YZ➁

ϕ

• where

ϕ ❃ R❼R❼L➁❼X,Y ➁➁❼R❼L➁❼Z,X➁,R❼L➁❼Y ,Z➁➁, ψ ❃ R❼R❼L➁❼Y ,Z➁➁❼R❼L➁❼Z,X➁,R❼L➁❼X,Y ➁➁, χ ❃ R❼R❼L➁❼Z,X➁➁❼R❼L➁❼Y ,Z➁,R❼L➁❼X,Y ➁➁ are defined, ➛α ❃ R❼L➁❼Z,X➁,β ❃ R❼L➁❼Y ,Z➁,γ ❃ R❼L➁❼X,Y ➁ by: ❨ ϕ❼α,β➁ α✏ ☎ β✏; ❨ ψ❼α,γ➁ γ✏ ✝❛ α✏; ❨ χ❼β,γ➁ β✏ ☎✂ γ✏, is a strong L-Galois triangle if and only if L is associative and commutative.

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L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Adjoint product Symmetry Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle

Proposition

Let L ❼L,,➋➁ be a symmetrical cdeo algebra. Then for any triple of sets ❼X,Y ,Z➁ the diagram ❼R❼L➁❼X,Y ➁,❚XY ➁

χ

• ❼R❼L➁❼Z,X➁,❚ZX➁

ψ

• ❼R❼L➁❼Y ,Z➁,❚YZ➁

ϕ

• where

ϕ ❃ R❼R❼L➁❼X,Y ➁➁❼R❼L➁❼Z,X➁,R❼L➁❼Y ,Z➁➁, ψ ❃ R❼R❼L➁❼Y ,Z➁➁❼R❼L➁❼Z,X➁,R❼L➁❼X,Y ➁➁, χ ❃ R❼R❼L➁❼Z,X➁➁❼R❼L➁❼Y ,Z➁,R❼L➁❼X,Y ➁➁ are defined, ➛α ❃ R❼L➁❼Z,X➁,β ❃ R❼L➁❼Y ,Z➁,γ ❃ R❼L➁❼X,Y ➁ by: ❨ ϕ❼α,β➁ α✏ ✝❛ β✏; ❨ ψ❼α,γ➁ γ✏ ✝❛ α✏; ❨ χ❼β,γ➁ β✏ ✝❛ γ✏, is a strong L-Galois triangle if and only if L is associative and commutative.

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