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

� ❼ ➁ � ☎ ➌ ❙ ❼ ➁➑ ✂ � ③ ❃ ❇ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Remark Preliminaries ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Galois connections Extended-order maps. If f Ú g, then: algebras Basic notions ▲ f preserves existing sups; Adjoint product Symmetry ▲ g preserves existing infs. Commutativity and associativity ▲ Let L be a complete lattice, M a poset and let f ✂ L � M be a L-relations Relation algebras map that preserves sups.Then the function MV-relation algebras Dedekind categories g ✂ M � L, y ③ � g ❼ y ➁ � ✝ ➌ x ❃ L ❙ f ❼ x ➁ ❇ y ➑ Extended-order algebras and L-relations is the unique right adjoint of f . L-Galois triangles ▲ Let L be a poset, M a complete lattice and let g ✂ M � L be Weak L-Galois triangle a map that preserves infs. Symmetrical L-Galois triangle Strong L-Galois triangle 4 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Remark Preliminaries ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Galois connections Extended-order maps. If f Ú g, then: algebras Basic notions ▲ f preserves existing sups; Adjoint product Symmetry ▲ g preserves existing infs. Commutativity and associativity ▲ Let L be a complete lattice, M a poset and let f ✂ L � M be a L-relations Relation algebras map that preserves sups.Then the function MV-relation algebras Dedekind categories g ✂ M � L, y ③ � g ❼ y ➁ � ✝ ➌ x ❃ L ❙ f ❼ x ➁ ❇ y ➑ Extended-order algebras and L-relations is the unique right adjoint of f . L-Galois triangles ▲ Let L be a poset, M a complete lattice and let g ✂ M � L be Weak L-Galois triangle a map that preserves infs.Then the function Symmetrical L-Galois triangle Strong L-Galois f ✂ L � M, x ③ � f ❼ x ➁ � ☎ ➌ y ❃ M ❙ x ❇ g ❼ y ➁➑ triangle is the unique left adjoint of g. 4 / 41

❼ ➁ � ✝ ➌ ❃ ❙ ❇ ❼ ➁➑ ✂ � ③ � � ✆ ▲ � ✆ � ✆ � ✆ ❼ ✝ ➁ � ☎ ❼ ➁ ❜ ▲ ✂ ▲ � ❼ ✝ ➁ � ☎ ❼ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Basic notions maps. Then 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 5 / 41

❼ ➁ � ✝ ➌ ❃ ❙ ❇ ❼ ➁➑ ✂ � ③ � � ✆ � ✆ ❼ ✝ ➁ � ☎ ❼ ➁ ❜ ▲ ✂ ▲ � ❼ ✝ ➁ � ☎ ❼ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Basic notions maps. Then Adjoint product Symmetry ▲ � g , f ✆ iff � f , g ✆ is; 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 5 / 41

❼ ➁ � ✝ ➌ ❃ ❙ ❇ ❼ ➁➑ ✂ � ③ � � ✆ ✂ ▲ � ❼ ✝ ➁ � ☎ ❼ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Basic notions maps. Then Adjoint product Symmetry ▲ � g , f ✆ iff � f , g ✆ is; Commutativity and associativity ▲ if � f , g ✆ and A ❜ L has a supremum, then f ❼ ✝ A ➁ � ☎ f ❼ A ➁ . 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 5 / 41

❼ ➁ � ✝ ➌ ❃ ❙ ❇ ❼ ➁➑ ✂ � ③ � � ✆ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Basic notions maps. Then Adjoint product Symmetry ▲ � g , f ✆ iff � f , g ✆ is; Commutativity and associativity ▲ if � f , g ✆ and A ❜ L has a supremum, then f ❼ ✝ A ➁ � ☎ f ❼ A ➁ . L-relations Relation algebras ▲ Let L be a complete lattice, M a poset and let f ✂ L � M be a MV-relation algebras Dedekind categories function such that f ❼ ✝ A ➁ � ☎ f ❼ A ➁ . Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 5 / 41

❼ ➁ � ✝ ➌ ❃ ❙ ❇ ❼ ➁➑ ✂ � ③ � � ✆ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Basic notions maps. Then Adjoint product Symmetry ▲ � g , f ✆ iff � f , g ✆ is; Commutativity and associativity ▲ if � f , g ✆ and A ❜ L has a supremum, then f ❼ ✝ A ➁ � ☎ f ❼ A ➁ . L-relations Relation algebras ▲ Let L be a complete lattice, M a poset and let f ✂ L � M be a MV-relation algebras Dedekind categories function such that f ❼ ✝ A ➁ � ☎ f ❼ A ➁ . Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 5 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ Let L and M be posets and let f ✂ L � M and g ✂ M � L be Basic notions maps. Then Adjoint product Symmetry ▲ � g , f ✆ iff � f , g ✆ is; Commutativity and associativity ▲ if � f , g ✆ and A ❜ L has a supremum, then f ❼ ✝ A ➁ � ☎ f ❼ A ➁ . L-relations Relation algebras ▲ Let L be a complete lattice, M a poset and let f ✂ L � M be a MV-relation algebras Dedekind categories function such that f ❼ ✝ A ➁ � ☎ f ❼ A ➁ .Then the function Extended-order algebras and g ✂ M � L , y ③ � g ❼ y ➁ � ✝ ➌ x ❃ L ❙ y ❇ f ❼ x ➁➑ L-relations L-Galois triangles is the unique function such that � f , g ✆ . Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 5 / 41

▲ ▲ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ C. Guido, P. Toto: Extended-order algebras , Journal of Basic notions Adjoint product Symmetry Applied Logic, 6 (4) (2008), 609-626. 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 6 / 41

▲ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ C. Guido, P. Toto: Extended-order algebras , Journal of Basic notions Adjoint product Symmetry Applied Logic, 6 (4) (2008), 609-626. Commutativity and associativity ▲ H. Rasiowa: An Algebraic Approach to Non-Classical Logics, L-relations Studies in Logics and the Foundations of Mathematics , Relation algebras MV-relation algebras vol.78, North-Holland, Amsterdam, 1974. Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 6 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras ▲ C. Guido, P. Toto: Extended-order algebras , Journal of Basic notions Adjoint product Symmetry Applied Logic, 6 (4) (2008), 609-626. Commutativity and associativity ▲ H. Rasiowa: An Algebraic Approach to Non-Classical Logics, L-relations Studies in Logics and the Foundations of Mathematics , Relation algebras MV-relation algebras vol.78, North-Holland, Amsterdam, 1974. Dedekind categories Extended-order ▲ M.E.D.S., C. Guido: Associativity, commutativity and algebras and L-relations symmetry in residuated structures , (submitted). L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 6 / 41

▲ ❼ ➁ � ➋ � ➋ ▲ ❼ ➁ � ➋ � ▲ ❼ ➁ � ➋ � ➋ ✟ � � � ▲ ❼ ➁ � ➋ � ➋ ✟ � ➋ � � � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras Let L be a non-empty set, � ✂ L ✕ L � L a binary operation and ➋ a Basic notions Adjoint product fixed element of L. The triple L � ❼ L , � , ➋ ➁ is a weak Symmetry Commutativity and extended-order algebra, shortly w-eo algebra, if 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 7 / 41

▲ ❼ ➁ � ➋ � ▲ ❼ ➁ � ➋ � ➋ ✟ � � � ▲ ❼ ➁ � ➋ � ➋ ✟ � ➋ � � � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras Let L be a non-empty set, � ✂ L ✕ L � L a binary operation and ➋ a Basic notions Adjoint product fixed element of L. The triple L � ❼ L , � , ➋ ➁ is a weak Symmetry Commutativity and extended-order algebra, shortly w-eo algebra, if associativity ▲ ❼ o 1 ➁ a � ➋ � ➋ (upper bound condition); 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 7 / 41

▲ ❼ ➁ � ➋ � ➋ ✟ � � � ▲ ❼ ➁ � ➋ � ➋ ✟ � ➋ � � � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras Let L be a non-empty set, � ✂ L ✕ L � L a binary operation and ➋ a Basic notions Adjoint product fixed element of L. The triple L � ❼ L , � , ➋ ➁ is a weak Symmetry Commutativity and extended-order algebra, shortly w-eo algebra, if associativity ▲ ❼ o 1 ➁ a � ➋ � ➋ (upper bound condition); L-relations Relation algebras ▲ ❼ o 2 ➁ a � a � ➋ (reflexivity condition); 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 7 / 41

▲ ❼ ➁ � ➋ � ➋ ✟ � ➋ � � � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras Let L be a non-empty set, � ✂ L ✕ L � L a binary operation and ➋ a Basic notions Adjoint product fixed element of L. The triple L � ❼ L , � , ➋ ➁ is a weak Symmetry Commutativity and extended-order algebra, shortly w-eo algebra, if associativity ▲ ❼ o 1 ➁ a � ➋ � ➋ (upper bound condition); L-relations Relation algebras ▲ ❼ o 2 ➁ a � a � ➋ (reflexivity condition); MV-relation algebras Dedekind categories Extended-order ▲ ❼ o 3 ➁ a � b � ➋ and b � a � ➋ ✟ a � b (antisymmetry algebras and L-relations condition); L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 7 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras Let L be a non-empty set, � ✂ L ✕ L � L a binary operation and ➋ a Basic notions Adjoint product fixed element of L. The triple L � ❼ L , � , ➋ ➁ is a weak Symmetry Commutativity and extended-order algebra, shortly w-eo algebra, if associativity ▲ ❼ o 1 ➁ a � ➋ � ➋ (upper bound condition); L-relations Relation algebras ▲ ❼ o 2 ➁ a � a � ➋ (reflexivity condition); MV-relation algebras Dedekind categories Extended-order ▲ ❼ o 3 ➁ a � b � ➋ and b � a � ➋ ✟ a � b (antisymmetry algebras and L-relations condition); L-Galois triangles Weak L-Galois ▲ ❼ o 4 ➁ a � b � ➋ and b � c � ➋ ✟ a � c � ➋ (weak transitivity triangle Symmetrical L-Galois condition). triangle Strong L-Galois triangle 7 / 41

▲ ❼ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ▲ ❼ ➐ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ▲ ❇ � � ➋ ❇ � ➋ ❼ ❇ ➁ ❼ ❇ ➁ ➋ ▲ ❼ � ➋ ➁ � ✂ ✕ � ➋ ✔ � ❇ � ❇ ❼ � ➋ ➁ ❼ ➁ ❼ ➁ ❼ ➁ L-relations and Galois triangles Proposition 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 8 / 41

▲ ❼ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ▲ ❼ ➐ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ❼ ❇ ➁ ➋ ▲ ❼ � ➋ ➁ � ✂ ✕ � ➋ ✔ � ❇ � ❇ ❼ � ➋ ➁ ❼ ➁ ❼ ➁ ❼ ➁ L-relations and Galois triangles Proposition M.Emilia Della Stella , Cosimo Guido ▲ The relation ❇ determined by the operation � , by means of Preliminaries the equivalence Galois connections Extended-order a ❇ b iff a � b � ➋ algebras Basic notions is an order relation in L. Moreover ➋ is a greatest element in Adjoint product Symmetry ❼ L , ❇ ➁ . 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 8 / 41

▲ ❼ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ▲ ❼ ➐ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ❼ � ➋ ➁ ❼ ➁ ❼ ➁ ❼ ➁ L-relations and Galois triangles Proposition M.Emilia Della Stella , Cosimo Guido ▲ The relation ❇ determined by the operation � , by means of Preliminaries the equivalence Galois connections Extended-order a ❇ b iff a � b � ➋ algebras Basic notions is an order relation in L. Moreover ➋ is a greatest element in Adjoint product Symmetry ❼ L , ❇ ➁ . Commutativity and associativity ▲ Conversely, if ❼ L , ❇ ➁ is a poset with a greatest element ➋ and L-relations Relation algebras � ✂ L ✕ L � L extends ❇ , i.e. a � b � ➋ ✔ a ❇ b, then ❼ L , � , ➋ ➁ MV-relation algebras Dedekind categories is a w-eo algebra. Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 8 / 41

▲ ❼ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ▲ ❼ ➐ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ❼ � ➋ ➁ ❼ ➁ ❼ ➁ ❼ ➁ L-relations and Galois triangles Proposition M.Emilia Della Stella , Cosimo Guido ▲ The relation ❇ determined by the operation � , by means of Preliminaries the equivalence Galois connections Extended-order a ❇ b iff a � b � ➋ algebras Basic notions is an order relation in L. Moreover ➋ is a greatest element in Adjoint product Symmetry ❼ L , ❇ ➁ . Commutativity and associativity ▲ Conversely, if ❼ L , ❇ ➁ is a poset with a greatest element ➋ and L-relations Relation algebras � ✂ L ✕ L � L extends ❇ , i.e. a � b � ➋ ✔ a ❇ b, then ❼ L , � , ➋ ➁ MV-relation algebras Dedekind categories is a w-eo algebra. Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 8 / 41

▲ ❼ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � ▲ ❼ ➐ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � L-relations and Galois triangles Proposition M.Emilia Della Stella , Cosimo Guido ▲ The relation ❇ determined by the operation � , by means of Preliminaries the equivalence Galois connections Extended-order a ❇ b iff a � b � ➋ algebras Basic notions is an order relation in L. Moreover ➋ is a greatest element in Adjoint product Symmetry ❼ L , ❇ ➁ . Commutativity and associativity ▲ Conversely, if ❼ L , ❇ ➁ is a poset with a greatest element ➋ and L-relations Relation algebras � ✂ L ✕ L � L extends ❇ , i.e. a � b � ➋ ✔ a ❇ b, then ❼ L , � , ➋ ➁ MV-relation algebras Dedekind categories is a w-eo algebra. Extended-order algebras and L-relations L-Galois triangles Definition Weak L-Galois triangle ❼ L , � , ➋ ➁ is an extended-order algebra, shortly eo algebra, if it Symmetrical L-Galois triangle satisfies the axioms ❼ o 1 ➁ , ❼ o 2 ➁ , ❼ o 3 ➁ and Strong L-Galois triangle 8 / 41

▲ ❼ ➐ ➁ � ➋ ✟ ❼ � ➁ � ❼ � ➁ � ➋ � L-relations and Galois triangles Proposition M.Emilia Della Stella , Cosimo Guido ▲ The relation ❇ determined by the operation � , by means of Preliminaries the equivalence Galois connections Extended-order a ❇ b iff a � b � ➋ algebras Basic notions is an order relation in L. Moreover ➋ is a greatest element in Adjoint product Symmetry ❼ L , ❇ ➁ . Commutativity and associativity ▲ Conversely, if ❼ L , ❇ ➁ is a poset with a greatest element ➋ and L-relations Relation algebras � ✂ L ✕ L � L extends ❇ , i.e. a � b � ➋ ✔ a ❇ b, then ❼ L , � , ➋ ➁ MV-relation algebras Dedekind categories is a w-eo algebra. Extended-order algebras and L-relations L-Galois triangles Definition Weak L-Galois triangle ❼ L , � , ➋ ➁ is an extended-order algebra, shortly eo algebra, if it Symmetrical L-Galois triangle satisfies the axioms ❼ o 1 ➁ , ❼ o 2 ➁ , ❼ o 3 ➁ and Strong L-Galois triangle ▲ ❼ o 5 ➁ a � b � ➋ ✟ ❼ c � a ➁ � ❼ c � b ➁ � ➋ (weak isotonic condition in the second variable); 8 / 41

L-relations and Galois triangles Proposition M.Emilia Della Stella , Cosimo Guido ▲ The relation ❇ determined by the operation � , by means of Preliminaries the equivalence Galois connections Extended-order a ❇ b iff a � b � ➋ algebras Basic notions is an order relation in L. Moreover ➋ is a greatest element in Adjoint product Symmetry ❼ L , ❇ ➁ . Commutativity and associativity ▲ Conversely, if ❼ L , ❇ ➁ is a poset with a greatest element ➋ and L-relations Relation algebras � ✂ L ✕ L � L extends ❇ , i.e. a � b � ➋ ✔ a ❇ b, then ❼ L , � , ➋ ➁ MV-relation algebras Dedekind categories is a w-eo algebra. Extended-order algebras and L-relations L-Galois triangles Definition Weak L-Galois triangle ❼ L , � , ➋ ➁ is an extended-order algebra, shortly eo algebra, if it Symmetrical L-Galois triangle satisfies the axioms ❼ o 1 ➁ , ❼ o 2 ➁ , ❼ o 3 ➁ and Strong L-Galois triangle ▲ ❼ o 5 ➁ a � b � ➋ ✟ ❼ c � a ➁ � ❼ c � b ➁ � ➋ (weak isotonic condition in the second variable); ▲ ❼ o ➐ 5 ➁ a � b � ➋ ✟ ❼ b � c ➁ � ❼ a � c ➁ � ➋ (weak antitonic condition in the first variable). 8 / 41

▲ ▲ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Any one of the algebras defined above is said to be complete if L Galois connections Extended-order with the order induced by � is a complete lattice. 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 9 / 41

▲ ▲ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Any one of the algebras defined above is said to be complete if L Galois connections Extended-order with the order induced by � is a complete lattice. algebras Basic notions Adjoint product Symmetry Commutativity and associativity From now, we consider only complete structures and we denote L-relations Relation algebras them with the obvious notation ( w -) ceo 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 9 / 41

▲ ▲ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Any one of the algebras defined above is said to be complete if L Galois connections Extended-order with the order induced by � is a complete lattice. algebras Basic notions Adjoint product Symmetry Commutativity and associativity From now, we consider only complete structures and we denote L-relations Relation algebras them with the obvious notation ( w -) ceo algebras. MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois Remark triangle Symmetrical L-Galois The completeness requirement is triangle Strong L-Galois triangle 9 / 41

▲ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Any one of the algebras defined above is said to be complete if L Galois connections Extended-order with the order induced by � is a complete lattice. algebras Basic notions Adjoint product Symmetry Commutativity and associativity From now, we consider only complete structures and we denote L-relations Relation algebras them with the obvious notation ( w -) ceo algebras. MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois Remark triangle Symmetrical L-Galois The completeness requirement is triangle Strong L-Galois ▲ not restrictive for eo algebras; triangle 9 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Any one of the algebras defined above is said to be complete if L Galois connections Extended-order with the order induced by � is a complete lattice. algebras Basic notions Adjoint product Symmetry Commutativity and associativity From now, we consider only complete structures and we denote L-relations Relation algebras them with the obvious notation ( w -) ceo algebras. MV-relation algebras Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois Remark triangle Symmetrical L-Galois The completeness requirement is triangle Strong L-Galois ▲ not restrictive for eo algebras; triangle ▲ restrictive for w-eo algebras. 9 / 41

▲ ❼ ➁ ✟ ❼ ➁ ▲ ❼ ➁ ✟ ❼ ➐ ➁ ▲ ❼ ➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼ ➐ ➁ ▲ ❼ � ➋ ➁ ❼ ➁ � ☎ ❼ � ➁ � ☎ ▲ ❼ � ➋ ➁ ❼ ➁ ❼ ✝ ➁ � � ☎ ❼ � ➁ ▲ ❼ � ➋ ➁ ❼ ➁ ✝ � ☎ ❼ ➁ � ☎ � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. 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 10 / 41

▲ ❼ ➁ ✟ ❼ ➁ ▲ ❼ ➁ ✟ ❼ ➐ ➁ ▲ ❼ ➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼ ➐ ➁ ▲ ❼ � ➋ ➁ ❼ ➁ ❼ ✝ ➁ � � ☎ ❼ � ➁ ▲ ❼ � ➋ ➁ ❼ ➁ ✝ � ☎ ❼ ➁ � ☎ � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. Galois connections Extended-order ▲ ❼ L , � , ➋ ➁ is right-distributive if it satisfies algebras Basic notions ❼ d r ➁ a � ☎ B � ☎ ❼ a � B ➁ . 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 10 / 41

▲ ❼ ➁ ✟ ❼ ➁ ▲ ❼ ➁ ✟ ❼ ➐ ➁ ▲ ❼ ➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼ ➐ ➁ ▲ ❼ � ➋ ➁ ❼ ➁ ✝ � ☎ ❼ ➁ � ☎ � L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. Galois connections Extended-order ▲ ❼ L , � , ➋ ➁ is right-distributive if it satisfies algebras Basic notions ❼ d r ➁ a � ☎ B � ☎ ❼ a � B ➁ . Adjoint product Symmetry ▲ ❼ L , � , ➋ ➁ is left-distributive if it satisfies Commutativity and associativity ❼ d l ➁ ❼ ✝ A ➁ � b � ☎ ❼ A � b ➁ . 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 10 / 41

▲ ❼ ➁ ✟ ❼ ➁ ▲ ❼ ➁ ✟ ❼ ➐ ➁ ▲ ❼ ➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼ ➐ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. Galois connections Extended-order ▲ ❼ L , � , ➋ ➁ is right-distributive if it satisfies algebras Basic notions ❼ d r ➁ a � ☎ B � ☎ ❼ a � B ➁ . Adjoint product Symmetry ▲ ❼ L , � , ➋ ➁ is left-distributive if it satisfies Commutativity and associativity ❼ d l ➁ ❼ ✝ A ➁ � b � ☎ ❼ A � b ➁ . L-relations Relation algebras ▲ ❼ L , � , ➋ ➁ is distributive if it satisfies MV-relation algebras Dedekind categories ❼ d ➁ ✝ A � ☎ B � ☎ ❼ A � B ➁ . Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 10 / 41

▲ ❼ ➁ ✟ ❼ ➐ ➁ ▲ ❼ ➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼ ➐ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. Galois connections Extended-order ▲ ❼ L , � , ➋ ➁ is right-distributive if it satisfies algebras Basic notions ❼ d r ➁ a � ☎ B � ☎ ❼ a � B ➁ . Adjoint product Symmetry ▲ ❼ L , � , ➋ ➁ is left-distributive if it satisfies Commutativity and associativity ❼ d l ➁ ❼ ✝ A ➁ � b � ☎ ❼ A � b ➁ . L-relations Relation algebras ▲ ❼ L , � , ➋ ➁ is distributive if it satisfies MV-relation algebras Dedekind categories ❼ d ➁ ✝ A � ☎ B � ☎ ❼ A � B ➁ . Extended-order algebras and L-relations L-Galois triangles Remark Weak L-Galois triangle ▲ ❼ d r ➁ ✟ ❼ o 5 ➁ ; Symmetrical L-Galois triangle Strong L-Galois triangle 10 / 41

▲ ❼ ➁ ❼ ➁ ✔ ❼ ➁ ✟ ❼ ➁ ❼ ➐ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. Galois connections Extended-order ▲ ❼ L , � , ➋ ➁ is right-distributive if it satisfies algebras Basic notions ❼ d r ➁ a � ☎ B � ☎ ❼ a � B ➁ . Adjoint product Symmetry ▲ ❼ L , � , ➋ ➁ is left-distributive if it satisfies Commutativity and associativity ❼ d l ➁ ❼ ✝ A ➁ � b � ☎ ❼ A � b ➁ . L-relations Relation algebras ▲ ❼ L , � , ➋ ➁ is distributive if it satisfies MV-relation algebras Dedekind categories ❼ d ➁ ✝ A � ☎ B � ☎ ❼ A � B ➁ . Extended-order algebras and L-relations L-Galois triangles Remark Weak L-Galois triangle ▲ ❼ d r ➁ ✟ ❼ o 5 ➁ ; Symmetrical L-Galois triangle Strong L-Galois ▲ ❼ d l ➁ ✟ ❼ o ➐ 5 ➁ ; triangle 10 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Definition Preliminaries Let ❼ L , � , ➋ ➁ be a w-ceo algebra. Galois connections Extended-order ▲ ❼ L , � , ➋ ➁ is right-distributive if it satisfies algebras Basic notions ❼ d r ➁ a � ☎ B � ☎ ❼ a � B ➁ . Adjoint product Symmetry ▲ ❼ L , � , ➋ ➁ is left-distributive if it satisfies Commutativity and associativity ❼ d l ➁ ❼ ✝ A ➁ � b � ☎ ❼ A � b ➁ . L-relations Relation algebras ▲ ❼ L , � , ➋ ➁ is distributive if it satisfies MV-relation algebras Dedekind categories ❼ d ➁ ✝ A � ☎ B � ☎ ❼ A � B ➁ . Extended-order algebras and L-relations L-Galois triangles Remark Weak L-Galois triangle ▲ ❼ d r ➁ ✟ ❼ o 5 ➁ ; Symmetrical L-Galois triangle Strong L-Galois ▲ ❼ d l ➁ ✟ ❼ o ➐ 5 ➁ ; triangle ▲ ❼ d r ➁ + ❼ d l ➁ ✔ ❼ d ➁ ✟ ❼ o 5 ➁ + ❼ o ➐ 5 ➁ . 10 / 41

▲ ❛ ❛ � ❇ ✔ ❇ � ▲ ✂ ❼ ➁ � ☎ � ✭ � ❼ ➁ ❛ ➋ � ▲ ❛ ➊ � ➊ ❛ � ➊ ▲ ❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① � ☎ ➌ ❃ ❙ � ➑ ❛ ❇ L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by 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 11 / 41

▲ ❛ ❛ � ❇ ✔ ❇ � ▲ ✂ ❼ ➁ � ☎ � ✭ � ❼ ➁ ❛ ➋ � ▲ ❛ ➊ � ➊ ❛ � ➊ ▲ ❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . 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 11 / 41

❛ ➋ � ▲ ❛ ➊ � ➊ ❛ � ➊ ▲ ❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① ▲ ✂ ❼ ➁ � ☎ � ✭ � ❼ ➁ L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. 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 11 / 41

❛ ➋ � ▲ ❛ ➊ � ➊ ❛ � ➊ ▲ ❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. L-relations Relation algebras ▲ The above Definition is justified by adjunction applied to the MV-relation algebras Dedekind categories function g a ✂ L � L, y ✭ g a ❼ y ➁ � a � y that preserves ☎ , Extended-order algebras and L-relations because the condition ❼ d r ➁ is assumed on L; L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 11 / 41

❛ ➋ � ▲ ❛ ➊ � ➊ ❛ � ➊ ▲ ❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. L-relations Relation algebras ▲ The above Definition is justified by adjunction applied to the MV-relation algebras Dedekind categories function g a ✂ L � L, y ✭ g a ❼ y ➁ � a � y that preserves ☎ , Extended-order algebras and L-relations because the condition ❼ d r ➁ is assumed on L; L-Galois triangles Weak L-Galois triangle Proposition Symmetrical L-Galois triangle Strong L-Galois triangle 11 / 41

❛ ➊ � ➊ ❛ � ➊ ▲ ❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. L-relations Relation algebras ▲ The above Definition is justified by adjunction applied to the MV-relation algebras Dedekind categories function g a ✂ L � L, y ✭ g a ❼ y ➁ � a � y that preserves ☎ , Extended-order algebras and L-relations because the condition ❼ d r ➁ is assumed on L; L-Galois triangles Weak L-Galois triangle Proposition Symmetrical L-Galois triangle Strong L-Galois triangle ▲ a ❛ ➋ � a; 11 / 41

❛ ❼ ✝ ➁ � ✝ ❼ ❛ ➁ ▲ ❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. L-relations Relation algebras ▲ The above Definition is justified by adjunction applied to the MV-relation algebras Dedekind categories function g a ✂ L � L, y ✭ g a ❼ y ➁ � a � y that preserves ☎ , Extended-order algebras and L-relations because the condition ❼ d r ➁ is assumed on L; L-Galois triangles Weak L-Galois triangle Proposition Symmetrical L-Galois triangle Strong L-Galois triangle ▲ a ❛ ➋ � a; ▲ a ❛ ➊ � ➊ ❛ a � ➊ ; 11 / 41

❛ ❼ ❛ ➁ ① ❼ ❛ ➁ ❛ ❛ ❛ ▲ ① L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. L-relations Relation algebras ▲ The above Definition is justified by adjunction applied to the MV-relation algebras Dedekind categories function g a ✂ L � L, y ✭ g a ❼ y ➁ � a � y that preserves ☎ , Extended-order algebras and L-relations because the condition ❼ d r ➁ is assumed on L; L-Galois triangles Weak L-Galois triangle Proposition Symmetrical L-Galois triangle Strong L-Galois triangle ▲ a ❛ ➋ � a; ▲ a ❛ ➊ � ➊ ❛ a � ➊ ; ▲ a ❛ ❼ ✝ B ➁ � ✝ ❼ a ❛ B ➁ ; 11 / 41

L-relations and Galois Definition triangles M.Emilia Della Stella , Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The adjoint Cosimo Guido product of L is the operation ❛ ✂ L ✕ L � L defined by Preliminaries Galois connections a ❛ b � ☎ ➌ t ❃ L ❙ b ❇ a � t ➑ . Extended-order algebras Basic notions Adjoint product Symmetry Remark Commutativity and associativity ▲ ❛ and � form an adjoint pair, hence x ❛ y ❇ z ✔ y ❇ x � z. L-relations Relation algebras ▲ The above Definition is justified by adjunction applied to the MV-relation algebras Dedekind categories function g a ✂ L � L, y ✭ g a ❼ y ➁ � a � y that preserves ☎ , Extended-order algebras and L-relations because the condition ❼ d r ➁ is assumed on L; L-Galois triangles Weak L-Galois triangle Proposition Symmetrical L-Galois triangle Strong L-Galois triangle ▲ a ❛ ➋ � a; ▲ a ❛ ➊ � ➊ ❛ a � ➊ ; ▲ a ❛ ❼ ✝ B ➁ � ✝ ❼ a ❛ B ➁ ; ▲ a ❛ b ① b ❛ a and a ❛ ❼ b ❛ c ➁ ① ❼ a ❛ b ➁ ❛ c, in general. 11 / 41

▲ � ➝ ▲ ❼ ➝ ➋ ➁ ❼ � ➋ ➁ ▲ ➜ ❼ ➝ ➋ ➁ ▲ ❇ ➝ ✔ ❇ � � ➝ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras A w-ceo algebra ❼ L , � , ➋ ➁ is symmetrical if 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 12 / 41

▲ � ➝ ▲ ❼ ➝ ➋ ➁ ❼ � ➋ ➁ ▲ ❇ ➝ ✔ ❇ � � ➝ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras A w-ceo algebra ❼ L , � , ➋ ➁ is symmetrical if Basic notions Adjoint product Symmetry ▲ ➜ ❼ L , ➝ , ➋ ➁ w-ceo algebra, with the same induced order; 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 12 / 41

▲ � ➝ ▲ ❼ ➝ ➋ ➁ ❼ � ➋ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras A w-ceo algebra ❼ L , � , ➋ ➁ is symmetrical if Basic notions Adjoint product Symmetry ▲ ➜ ❼ L , ➝ , ➋ ➁ w-ceo algebra, with the same induced order; Commutativity and associativity ▲ y ❇ x ➝ z ✔ x ❇ y � z (i.e.[ � , ➝ ]). 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 12 / 41

▲ � ➝ ▲ ❼ ➝ ➋ ➁ ❼ � ➋ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras A w-ceo algebra ❼ L , � , ➋ ➁ is symmetrical if Basic notions Adjoint product Symmetry ▲ ➜ ❼ L , ➝ , ➋ ➁ w-ceo algebra, with the same induced order; Commutativity and associativity ▲ y ❇ x ➝ z ✔ x ❇ y � z (i.e.[ � , ➝ ]). L-relations Relation algebras MV-relation algebras Remark Dedekind categories Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 12 / 41

▲ ❼ ➝ ➋ ➁ ❼ � ➋ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras A w-ceo algebra ❼ L , � , ➋ ➁ is symmetrical if Basic notions Adjoint product Symmetry ▲ ➜ ❼ L , ➝ , ➋ ➁ w-ceo algebra, with the same induced order; Commutativity and associativity ▲ y ❇ x ➝ z ✔ x ❇ y � z (i.e.[ � , ➝ ]). L-relations Relation algebras MV-relation algebras Remark Dedekind categories Extended-order algebras and ▲ Since � and ➝ form a Galois pair, each of them is uniquely L-relations L-Galois triangles determined by the other one. Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 12 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras A w-ceo algebra ❼ L , � , ➋ ➁ is symmetrical if Basic notions Adjoint product Symmetry ▲ ➜ ❼ L , ➝ , ➋ ➁ w-ceo algebra, with the same induced order; Commutativity and associativity ▲ y ❇ x ➝ z ✔ x ❇ y � z (i.e.[ � , ➝ ]). L-relations Relation algebras MV-relation algebras Remark Dedekind categories Extended-order algebras and ▲ Since � and ➝ form a Galois pair, each of them is uniquely L-relations L-Galois triangles determined by the other one. Weak L-Galois triangle ▲ ❼ L , ➝ , ➋ ➁ is symmetrical iff ❼ L , � , ➋ ➁ is symmetrical. Symmetrical L-Galois triangle Strong L-Galois triangle 12 / 41

❼ ➝ ➋ ➁ ❛ ▲ ❛ ❛ ❛ � ❛ ▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ ▲ ❼ � ➋ ➁ ▲ ❼ ➝ ➋ ➁ ▲ ➋ ❛ � ▲ ❼ ✝ ➁ ❛ � ✝ ❼ ❛ ➁ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition 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 13 / 41

❼ ➝ ➋ ➁ ❛ ▲ ❛ ❛ ❛ � ❛ ▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ ▲ ❼ ➝ ➋ ➁ ▲ ➋ ❛ � ▲ ❼ ✝ ➁ ❛ � ✝ ❼ ❛ ➁ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; 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 13 / 41

❼ ➝ ➋ ➁ ❛ ▲ ❛ ❛ ❛ � ❛ ▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ ▲ ➋ ❛ � ▲ ❼ ✝ ➁ ❛ � ✝ ❼ ❛ ➁ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; 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 13 / 41

❼ ➝ ➋ ➁ ❛ ▲ ❛ ❛ ❛ � ❛ ▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ ▲ ❼ ✝ ➁ ❛ � ✝ ❼ ❛ ➁ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; 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 13 / 41

❼ ➝ ➋ ➁ ❛ ▲ ❛ ❛ ❛ � ❛ ▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; Commutativity and associativity ▲ ❼ ✝ B ➁ ❛ a � ✝ ❼ B ❛ a ➁ . 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 13 / 41

❼ ➝ ➋ ➁ ❛ ▲ ❛ ❛ ❛ � ❛ ▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; Commutativity and associativity ▲ ❼ ✝ B ➁ ❛ a � ✝ ❼ B ❛ a ➁ . L-relations Relation algebras MV-relation algebras Dedekind categories Remark Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 13 / 41

▲ ❛ ❛ ➝ ❇ ➝ ✔ ❇ ▲ ➋ ❛ ① ▲ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; Commutativity and associativity ▲ ❼ ✝ B ➁ ❛ a � ✝ ❼ B ❛ a ➁ . L-relations Relation algebras MV-relation algebras Dedekind categories Remark Extended-order algebras and L-relations ▲ The adjoint product ˜ ❛ of the cdeo algebra ❼ L , ➝ , ➋ ➁ is the L-Galois triangles opposite ❛ op of ❛ , i. e. a ˜ ❛ b � b ❛ a. Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 13 / 41

▲ ➋ ❛ ① ▲ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; Commutativity and associativity ▲ ❼ ✝ B ➁ ❛ a � ✝ ❼ B ❛ a ➁ . L-relations Relation algebras MV-relation algebras Dedekind categories Remark Extended-order algebras and L-relations ▲ The adjoint product ˜ ❛ of the cdeo algebra ❼ L , ➝ , ➋ ➁ is the L-Galois triangles opposite ❛ op of ❛ , i. e. a ˜ ❛ b � b ❛ a. Weak L-Galois triangle Symmetrical L-Galois ▲ ❛ and ➝ are related by the equivalence a ❇ b ➝ c ✔ a ❛ b ❇ c. triangle Strong L-Galois triangle 13 / 41

▲ Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; Commutativity and associativity ▲ ❼ ✝ B ➁ ❛ a � ✝ ❼ B ❛ a ➁ . L-relations Relation algebras MV-relation algebras Dedekind categories Remark Extended-order algebras and L-relations ▲ The adjoint product ˜ ❛ of the cdeo algebra ❼ L , ➝ , ➋ ➁ is the L-Galois triangles opposite ❛ op of ❛ , i. e. a ˜ ❛ b � b ❛ a. Weak L-Galois triangle Symmetrical L-Galois ▲ ❛ and ➝ are related by the equivalence a ❇ b ➝ c ✔ a ❛ b ❇ c. triangle Strong L-Galois ▲ The cdeo algebras need not to be symmetrical. In fact in the triangle cdeo algebras ➋ ❛ a ① a, in general. 13 / 41

Under right-distributivity and symmetry assumptions we have L-relations and Galois triangles further properties. M.Emilia Della Stella , Cosimo Guido Proposition Preliminaries Galois connections ▲ ❼ L , � , ➋ ➁ is a cdeo algebra; Extended-order algebras ▲ ❼ L , ➝ , ➋ ➁ is a cdeo algebra; Basic notions Adjoint product Symmetry ▲ ➋ ❛ a � a; Commutativity and associativity ▲ ❼ ✝ B ➁ ❛ a � ✝ ❼ B ❛ a ➁ . L-relations Relation algebras MV-relation algebras Dedekind categories Remark Extended-order algebras and L-relations ▲ The adjoint product ˜ ❛ of the cdeo algebra ❼ L , ➝ , ➋ ➁ is the L-Galois triangles opposite ❛ op of ❛ , i. e. a ˜ ❛ b � b ❛ a. Weak L-Galois triangle Symmetrical L-Galois ▲ ❛ and ➝ are related by the equivalence a ❇ b ➝ c ✔ a ❛ b ❇ c. triangle Strong L-Galois ▲ The cdeo algebras need not to be symmetrical. In fact in the triangle 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

▲ ❼ � ➋ ➁ ❛ ▲ ▲ ❼ � ➋ ➁ ➝ � ❛ � ❛ ❼ � ➋ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Definition Galois connections Extended-order A w-ceo algebra ❼ L , � , ➋ ➁ is commutative iff algebras Basic notions ❼ c ➁ a � ❼ b � c ➁ � ➋ ✔ b � ❼ a � c ➁ � ➋ (weak exchange Adjoint product condition). 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 14 / 41

▲ ❼ � ➋ ➁ ❛ ▲ ▲ ❼ � ➋ ➁ ➝ � ❛ � ❛ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Definition Galois connections Extended-order A w-ceo algebra ❼ L , � , ➋ ➁ is commutative iff algebras Basic notions ❼ c ➁ a � ❼ b � c ➁ � ➋ ✔ b � ❼ a � c ➁ � ➋ (weak exchange Adjoint product condition). Symmetry Commutativity and associativity Proposition L-relations Relation algebras Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The followings MV-relation algebras Dedekind categories are equivalent: Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 14 / 41

❛ ▲ ▲ ❼ � ➋ ➁ ➝ � ❛ � ❛ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Definition Galois connections Extended-order A w-ceo algebra ❼ L , � , ➋ ➁ is commutative iff algebras Basic notions ❼ c ➁ a � ❼ b � c ➁ � ➋ ✔ b � ❼ a � c ➁ � ➋ (weak exchange Adjoint product condition). Symmetry Commutativity and associativity Proposition L-relations Relation algebras Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The followings MV-relation algebras Dedekind categories are equivalent: Extended-order algebras and L-relations ▲ ❼ L , � , ➋ ➁ is commutative; L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 14 / 41

▲ ❼ � ➋ ➁ ➝ � ❛ � ❛ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Definition Galois connections Extended-order A w-ceo algebra ❼ L , � , ➋ ➁ is commutative iff algebras Basic notions ❼ c ➁ a � ❼ b � c ➁ � ➋ ✔ b � ❼ a � c ➁ � ➋ (weak exchange Adjoint product condition). Symmetry Commutativity and associativity Proposition L-relations Relation algebras Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The followings MV-relation algebras Dedekind categories are equivalent: Extended-order algebras and L-relations ▲ ❼ L , � , ➋ ➁ is commutative; L-Galois triangles ▲ the adjoint product ❛ is commutative; Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 14 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Definition Galois connections Extended-order A w-ceo algebra ❼ L , � , ➋ ➁ is commutative iff algebras Basic notions ❼ c ➁ a � ❼ b � c ➁ � ➋ ✔ b � ❼ a � c ➁ � ➋ (weak exchange Adjoint product condition). Symmetry Commutativity and associativity Proposition L-relations Relation algebras Let ❼ L , � , ➋ ➁ be a right-distributive w-ceo algebra. The followings MV-relation algebras Dedekind categories are equivalent: Extended-order algebras and L-relations ▲ ❼ L , � , ➋ ➁ is commutative; L-Galois triangles ▲ the adjoint product ❛ is commutative; Weak L-Galois triangle Symmetrical L-Galois ▲ ❼ L , � , ➋ ➁ is symmetrical and ➝ coincides with � (and, of triangle Strong L-Galois course, ˜ ❛ � ❛ ). triangle 14 / 41

▲ ❼ � ➋ ➁ ▲ ❼ ➝ ➋ ➁ ➝ ❼ ➝ ➁ � ❼ ❛ ➁ ➝ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ❼ � ➋ ➁ ▲ ▲ ❛ ▲ ▲ ❼ ➁ � ❼ � ➁ � ❼ ❛ ➁ � ❼ � ➋ ➁ ▲ ❼ � ➋ ➁ ▲ 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 15 / 41

▲ ❼ � ➋ ➁ ▲ ❼ ➝ ➋ ➁ ➝ ❼ ➝ ➁ � ❼ ❛ ➁ ➝ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ❼ � ➋ ➁ ▲ ▲ ❛ ▲ ▲ ❼ ➁ � ❼ � ➁ � ❼ ❛ ➁ � ❼ � ➋ ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition 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 15 / 41

▲ ❼ � ➋ ➁ ▲ ❼ ➝ ➋ ➁ ➝ ❼ ➝ ➁ � ❼ ❛ ➁ ➝ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ❼ � ➋ ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) 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 15 / 41

▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ▲ ❼ � ➋ ➁ ▲ ❼ ➝ ➋ ➁ ➝ ❼ ➝ ➁ � ❼ ❛ ➁ ➝ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: 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 15 / 41

▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ▲ ❼ ➝ ➋ ➁ ➝ ❼ ➝ ➁ � ❼ ❛ ➁ ➝ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 15 / 41

▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ➝ ❼ ➝ ➁ � ❼ ❛ ➁ ➝ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and ▲ ❼ L , ➝ , ➋ ➁ is associative. L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 15 / 41

▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ➝ ❼ � ➁ � � ❼ ➝ ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and ▲ ❼ L , ➝ , ➋ ➁ is associative. L-relations ▲ a ➝ ❼ b ➝ c ➁ � ❼ a ❛ b ➁ ➝ c. (strong adjuction) L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 15 / 41

▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ ❼ � ➋ ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and ▲ ❼ L , ➝ , ➋ ➁ is associative. L-relations ▲ a ➝ ❼ b ➝ c ➁ � ❼ a ❛ b ➁ ➝ c. (strong adjuction) L-Galois triangles Weak L-Galois ▲ a ➝ ❼ b � c ➁ � b � ❼ a ➝ c ➁ . (strong Galois connection) triangle Symmetrical L-Galois triangle Strong L-Galois triangle 15 / 41

▲ ❼ � ➋ ➁ � ❼ � ➁ � � ❼ � ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and ▲ ❼ L , ➝ , ➋ ➁ is associative. L-relations ▲ a ➝ ❼ b ➝ c ➁ � ❼ a ❛ b ➁ ➝ c. (strong adjuction) L-Galois triangles Weak L-Galois ▲ a ➝ ❼ b � c ➁ � b � ❼ a ➝ c ➁ . (strong Galois connection) triangle Symmetrical L-Galois ▲ If ❼ L , � , ➋ ➁ is a symmetrical and commutative cdeo algebra, triangle Strong L-Galois then the following are equivalent: triangle 15 / 41

� ❼ � ➁ � � ❼ � ➁ ▲ L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and ▲ ❼ L , ➝ , ➋ ➁ is associative. L-relations ▲ a ➝ ❼ b ➝ c ➁ � ❼ a ❛ b ➁ ➝ c. (strong adjuction) L-Galois triangles Weak L-Galois ▲ a ➝ ❼ b � c ➁ � b � ❼ a ➝ c ➁ . (strong Galois connection) triangle Symmetrical L-Galois ▲ If ❼ L , � , ➋ ➁ is a symmetrical and commutative cdeo algebra, triangle Strong L-Galois then the following are equivalent: triangle ▲ ❼ L , � , ➋ ➁ is associative. 15 / 41

L-relations and Galois triangles M.Emilia Della Stella , Proposition Cosimo Guido Preliminaries ▲ If ❼ L , � , ➋ ➁ is a right-distributive w-ceo algebra, the following Galois connections are equivalent: Extended-order algebras ▲ L is associative; Basic notions Adjoint product ▲ the adjoint product ❛ is associative; Symmetry Commutativity and ▲ ❼ a ➁ a � ❼ b � c ➁ � ❼ b ❛ a ➁ � c. (strong adjunction) associativity ▲ If ❼ L , � , ➋ ➁ is a symmetrical cdeo algebra, the following are L-relations Relation algebras equivalent: MV-relation algebras Dedekind categories ▲ ❼ L , � , ➋ ➁ is associative. Extended-order algebras and ▲ ❼ L , ➝ , ➋ ➁ is associative. L-relations ▲ a ➝ ❼ b ➝ c ➁ � ❼ a ❛ b ➁ ➝ c. (strong adjuction) L-Galois triangles Weak L-Galois ▲ a ➝ ❼ b � c ➁ � b � ❼ a ➝ c ➁ . (strong Galois connection) triangle Symmetrical L-Galois ▲ If ❼ L , � , ➋ ➁ is a symmetrical and commutative cdeo algebra, triangle Strong L-Galois then the following are equivalent: triangle ▲ ❼ L , � , ➋ ➁ is associative. ▲ a � ❼ b � c ➁ � b � ❼ a � c ➁ . (strong exchange condition) 15 / 41

❼ ✲ ✱ ✏ ❳ ✘ ➁ ❼ ➁ ▲ ❼ ✲ ✱ ✏ ➁ ▲ ❼ ➁ ❳ ✔ ❼ ✘ ❳ ➁ ✱ ▲ ❼ ❳ ➁ ✱ ✔ ❼ ❳ ✘ ➁ ✱ � � � ❼ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido B. J´ onsson, A. Tarski: Representation problems for relation Preliminaries algebras , Bull. Amer. Math. Soc. 54 (1948), 79-80. 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 16 / 41

❼ ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido B. J´ onsson, A. Tarski: Representation problems for relation Preliminaries algebras , Bull. Amer. Math. Soc. 54 (1948), 79-80. Galois connections Extended-order algebras Definition Basic notions A (classical) relation algebra is a structure ❼ A , ✲ , ✱ , ✏ , 0 , 1 , ❳ , ✘ , ∆ ➁ Adjoint product Symmetry of type ❼ 2 , 2 , 1 , 0 , 0 , 2 , 1 , 0 ➁ such that: Commutativity and associativity ▲ ❼ A , ✲ , ✱ , ✏ , 0 , 1 ➁ is a Boolean algebra; L-relations Relation algebras ▲ ❼ A , ❳ , ∆ ➁ is a monoid; MV-relation algebras Dedekind categories ▲ ❼ x ❳ y ➁ ✱ z � 0 ✔ ❼ x ✘ ❳ z ➁ ✱ y � 0 ✔ ❼ z ❳ y ✘ ➁ ✱ x � 0 (cycle Extended-order algebras and L-relations law). L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 16 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido B. J´ onsson, A. Tarski: Representation problems for relation Preliminaries algebras , Bull. Amer. Math. Soc. 54 (1948), 79-80. Galois connections Extended-order algebras Definition Basic notions A (classical) relation algebra is a structure ❼ A , ✲ , ✱ , ✏ , 0 , 1 , ❳ , ✘ , ∆ ➁ Adjoint product Symmetry of type ❼ 2 , 2 , 1 , 0 , 0 , 2 , 1 , 0 ➁ such that: Commutativity and associativity ▲ ❼ A , ✲ , ✱ , ✏ , 0 , 1 ➁ is a Boolean algebra; L-relations Relation algebras ▲ ❼ A , ❳ , ∆ ➁ is a monoid; MV-relation algebras Dedekind categories ▲ ❼ x ❳ y ➁ ✱ z � 0 ✔ ❼ x ✘ ❳ z ➁ ✱ y � 0 ✔ ❼ z ❳ y ✘ ➁ ✱ x � 0 (cycle Extended-order algebras and L-relations law). L-Galois triangles Weak L-Galois triangle Example Symmetrical L-Galois triangle Strong L-Galois Rel ❼ X ➁ , the algebra of classical binary relations on a set X, is the triangle standard example of relation algebra. 16 / 41

❵ ❜ ✏ ❼ ❳ ✘ ➁ ❼ ➁ ▲ ❼ ❵ ❜ ✏ ➁ ▲ ❼ ➁ ❳ ✔ ❼ ✘ ❳ ➁ ❜ ▲ ❼ ❳ ➁ ❜ ✔ ❼ ❳ ✘ ➁ ❜ � � � ✕ ❼ ➁ � ❼� ✆ ❵ ❜ ✏ ❳ ✘ ➁ � ✆ ▲ ✆ ❵ ❜ ✏ ➁ ▲ ❵ ❜ ✏ ❼� ▲ ❘ ❳ ❙ ❼ ➁ � ✝ ❃ ❘ ❼ ➁ ❜ ❙ ❼ ➁ ▲ ❘ ✘ ❼ ➁ � ❘ ❼ ➁ ❼ ➁ � ❼ ➁ � ▲ � ① L-relations and Galois A. Popescu: Many-valued relation algebras , Algebra Universalis 53 triangles (2005), 73-108. 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 17 / 41

✕ ❼ ➁ � ❼� ✆ ❵ ❜ ✏ ❳ ✘ ➁ � ✆ ▲ ✆ ❵ ❜ ✏ ➁ ▲ ❵ ❜ ✏ ❼� ▲ ❘ ❳ ❙ ❼ ➁ � ✝ ❃ ❘ ❼ ➁ ❜ ❙ ❼ ➁ ▲ ❘ ✘ ❼ ➁ � ❘ ❼ ➁ ❼ ➁ � ❼ ➁ � ▲ � ① L-relations and Galois A. Popescu: Many-valued relation algebras , Algebra Universalis 53 triangles (2005), 73-108. M.Emilia Della Stella , Cosimo Guido Definition Preliminaries An MV-relation algebra is a structure ❼ A , ❵ , ❜ , ✏ , 0 , 1 , ❳ , ✘ , ∆ ➁ of Galois connections Extended-order type ❼ 2 , 2 , 1 , 0 , 0 , 2 , 1 , 0 ➁ such that: algebras Basic notions ▲ ❼ A , ❵ , ❜ , ✏ , 0 , 1 ➁ is an MV-algebra; Adjoint product Symmetry Commutativity and ▲ ❼ A , ❳ , ∆ ➁ is a monoid; associativity ▲ ❼ x ❳ y ➁ ❜ z � 0 ✔ ❼ x ✘ ❳ z ➁ ❜ y � 0 ✔ ❼ z ❳ y ✘ ➁ ❜ x � 0 . 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 17 / 41

L-relations and Galois A. Popescu: Many-valued relation algebras , Algebra Universalis 53 triangles (2005), 73-108. M.Emilia Della Stella , Cosimo Guido Definition Preliminaries An MV-relation algebra is a structure ❼ A , ❵ , ❜ , ✏ , 0 , 1 , ❳ , ✘ , ∆ ➁ of Galois connections Extended-order type ❼ 2 , 2 , 1 , 0 , 0 , 2 , 1 , 0 ➁ such that: algebras Basic notions ▲ ❼ A , ❵ , ❜ , ✏ , 0 , 1 ➁ is an MV-algebra; Adjoint product Symmetry Commutativity and ▲ ❼ A , ❳ , ∆ ➁ is a monoid; associativity ▲ ❼ x ❳ y ➁ ❜ z � 0 ✔ ❼ x ✘ ❳ z ➁ ❜ y � 0 ✔ ❼ z ❳ y ✘ ➁ ❜ x � 0 . L-relations Relation algebras MV-relation algebras Dedekind categories Extended-order Example algebras and L-relations MVRel ❼ X ➁ � ❼� 0 , 1 ✆ X ✕ X , ❵ , ❜ , ✏ , 0 , 1 , ❳ , ✘ , ∆ ➁ is the classical L-Galois triangles algebra of binary � 0 , 1 ✆ -relations on a set X where: Weak L-Galois triangle Symmetrical L-Galois ▲ 0 and 1 are the constant relations; triangle Strong L-Galois ▲ ❵ , ❜ and ✏ are the pointwise operation from ❼� 0 , 1 ✆ , ❵ , ❜ , ✏ ➁ ; triangle ▲ ❘ ❳ ❙ ❼ x , y ➁ � ✝ z ❃ X ❘ ❼ x , z ➁ ❜ ❙ ❼ z , y ➁ ; ▲ ❘ ✘ ❼ x , y ➁ � ❘ ❼ y , x ➁ ; ▲ ∆ ❼ x , y ➁ � 1 if x � y and ∆ ❼ x , y ➁ � 0 if x ① y. 17 / 41

❉ � ▲ ❉ ❼ ➁ � ❼ ❉ ❼ ➁ ❜ ✽ ✾ ✟ ➁ ➞ ❼ ❉ ➁ ➛ ❃ ❜ � ✾ � ✽ ▲ ▲ ✟ ❜ ✾ ❜ ✟ ➞ ▲ ✂ ❉ ❼ ➁ � ❉ ❼ ➁ ▲ ➐ ✂ ➛ ➛ ✂ ❍ ❍ ▲ ❼ ➁ � � � ▲ ❼ ➁ � ➐ ➐ ❜ ❜ ▲ L-relations and Galois triangles M.Emilia Della Stella , H. Furusawa, Y. Kawahara, M. Winter: Dedekind Categories with Cosimo Guido Cutoff Operators , Fuzzy Sets Syst. (article in press). 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 18 / 41

L-relations and Galois triangles M.Emilia Della Stella , H. Furusawa, Y. Kawahara, M. Winter: Dedekind Categories with Cosimo Guido Cutoff Operators , Fuzzy Sets Syst. (article in press). Preliminaries Galois connections Definition Extended-order algebras A Dedekind category ❉ is a category with composition � such that: Basic notions Adjoint product Symmetry Commutativity and ▲ ❉ ❼ X , Y ➁ � ❼ ❉ ❼ X , Y ➁ , ❜ , ✽ , ✾ , ✟ , 0 XY , ➞ XY ➁ is an Heyting associativity algebra, ➛ X , Y ❃ Obj ❼ ❉ ➁ ,where: L-relations Relation algebras ▲ α ❜ β iff α � α ✾ β iff β � α ✽ β ; MV-relation algebras Dedekind categories ▲ α ✟ β is the relative pseudo-complement of α relative to β Extended-order algebras and i.e. γ ❜ α ✟ β iff α ✾ γ ❜ β ; L-relations ▲ 0 XY e ➞ XY are the least and the greatest element. L-Galois triangles Weak L-Galois ▲ There exists a converse operation # ✂ ❉ ❼ X , Y ➁ � ❉ ❼ Y , X ➁ triangle Symmetrical L-Galois such that ➛ α,α ➐ ✂ X ❍ Y , ➛ β ✂ Y ❍ Z: triangle Strong L-Galois ▲ ❼ α � β ➁ # � β # � α # ; triangle ▲ ❼ α # ➁ # � α ; ▲ if α ❜ α ➐ , then α # ❜ α ➐ # . 18 / 41

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 L-relations and Galois triangles M.Emilia Della Stella , ▲ ➛ α ✂ X ❍ Y ,β ✂ Y ❍ Z ,γ ✂ X ❍ Z : Cosimo Guido α � β ✾ γ ❜ α � ❼ β ✾ α # � γ ➁ . ❼ modular law ➁ Preliminaries Galois connections ▲ ➛ α ✂ X ❍ Y ,β ✂ Y ❍ Z the residual composition Extended-order algebras α ❭ β ✂ X ❍ Z is a morphism such that ➛ δ ✂ X ❍ Z , Basic notions δ ❜ α ❭ β iff α # � δ ❜ β . 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 19 / 41

L-relations and Galois triangles M.Emilia Della Stella , ▲ ➛ α ✂ X ❍ Y ,β ✂ Y ❍ Z ,γ ✂ X ❍ Z : Cosimo Guido α � β ✾ γ ❜ α � ❼ β ✾ α # � γ ➁ . ❼ modular law ➁ Preliminaries Galois connections ▲ ➛ α ✂ X ❍ Y ,β ✂ Y ❍ Z the residual composition Extended-order algebras α ❭ β ✂ X ❍ Z is a morphism such that ➛ δ ✂ X ❍ Z , Basic notions δ ❜ α ❭ β iff α # � δ ❜ β . Adjoint product Symmetry Commutativity and associativity Example L-relations Relation algebras Rel ❼ L ➁❼ X , Y ➁ � ❼ Rel ❼ L ➁❼ X , Y ➁ , ❜ , ✽ , ✾ , ✟ , 0 XY , ➞ XY ➁ is the MV-relation algebras Dedekind categories Dedekind category of binary heterogeneous L-relations taking Extended-order algebras and value in an Heyting algebra ❼ L , ✱ , ✲ , 0 , 1 , � ➁ , where: L-relations L-Galois triangles ▲ ❜ , ✽ , ✾ and ✟ are pointwise induced by ✲ , ✱ , ❇ and � of L; Weak L-Galois triangle ▲ ❘ # ❼ y , x ➁ � ❘ ❼ x , y ➁ ; Symmetrical L-Galois triangle ▲ ❘ � ❙ ❼ x , z ➁ � ✝ y ❃ Y ❘ ❼ x , y ➁ ✱ ❙ ❼ y , z ➁ ; Strong L-Galois triangle ▲ 0 XY � 0 and ➞ XY � 1 ; ▲ ❘ ❭ ❙ ❼ x , z ➁ � ☎ y ❃ Y ❘ ❼ x , y ➁ � ❙ ❼ y , z ➁ . 19 / 41

L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Extended-order algebras Basic notions Let L � ❼ L , � , ➋ ➁ be a w - ceo algebra and consider ➛ X , Y ❃ ❙ Set ❙ Adjoint product Symmetry the set R ❼ L ➁❼ X , Y ➁ of L -relations ❘ ✂ X ✕ Y � L ✁ X ❍ Y . Commutativity and associativity L-relations Relation algebras MV-relation algebras Dedekind categories What about R ❼ L ➁❼ X , Y ➁ ? Extended-order algebras and L-relations L-Galois triangles Weak L-Galois triangle Symmetrical L-Galois triangle Strong L-Galois triangle 20 / 41

▲ ã ã ❼ ➁ � ➋ ✂ ❍ ➐ ❃ ▲ ■ ■ ❼ ➁ � ➋ ■ ❼ ➐ ➁ � ➊ ➛ ① ✂ ❍ ▲ ❘ ❇ ❘ ➐ ✔ ❘ ❼ ➁ ❇ ❘ ➐ ❼ ➁ ▲ ❘ � ❘ ➐ ✂ ❘ � ❘ ➐ ❼ ➁ � ❘ ❼ ➁ � ❘ ➐ ❼ ➁ ❍ ❼ ➁❼ ➁ � ❼ ❼ ➁❼ ➁ � ã ➁ L-relations and Galois triangles M.Emilia Della Stella , Cosimo Guido Preliminaries Galois connections Definition Extended-order algebras ▲ á XY ✂ X ❍ Y : á XY ❼ x , y ➁ � ➊ ; 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 21 / 41