Bounded commutative residuated lattices with a retraction term. - PowerPoint PPT Presentation

Bounded commutative residuated lattices with a retraction term. Manuela Busaniche Instituto de Matem´ atica Aplicada del Litoral Santa Fe, Argentina BLAST 2018 Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence To the memory of Roberto Cignoli The ideas of this talk are based on joint works with Roberto Cignoli, Miguel Marcos and Sara Ugolini. Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Residuated lattices An integral and commutative residuated lattice is an algebra A = � A , ∗ , → , ∨ , ∧ , ⊤� such that Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Residuated lattices An integral and commutative residuated lattice is an algebra A = � A , ∗ , → , ∨ , ∧ , ⊤� such that � A , ∗ , ⊤� is a commutative monoid, Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Residuated lattices An integral and commutative residuated lattice is an algebra A = � A , ∗ , → , ∨ , ∧ , ⊤� such that � A , ∗ , ⊤� is a commutative monoid, L ( A ) = � A , ∨ , ∧ , ⊤� is a lattice with greatest element ⊤ , Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Residuated lattices An integral and commutative residuated lattice is an algebra A = � A , ∗ , → , ∨ , ∧ , ⊤� such that � A , ∗ , ⊤� is a commutative monoid, L ( A ) = � A , ∨ , ∧ , ⊤� is a lattice with greatest element ⊤ , the following residuation condition holds: x ∗ y ≤ z iff x ≤ y → z (1) Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Bounded residuated lattices A bounded residuated lattice is an algebra A = � A , ∗ , → , ∨ , ∧ , ⊤ , ⊥� such that � A , ∗ , → , ∨ , ∧ , ⊤� is a residuated lattice, and ⊥ is the smallest element of the lattice L ( A ) . Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Famous bounded residuated lattices Boolean algebras Heyting algebras MV-algebras BL-algebras MTL-algebras NM-algebras Nelson residuated lattices Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Historical background Montagna, F. and Ugolini, S., A categorical equivalence for product algebras , Studia Logica 103 (2015), 345-373. Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Historical background Montagna, F. and Ugolini, S., A categorical equivalence for product algebras , Studia Logica 103 (2015), 345-373. Chen, C. C. and Gr¨ atzer, G., Stone Lattices. I: Construction Theorems, Canad. J. Math. 21 (1969), 884–994. Katriˇ n´ ak, T., A new proof of the construction theorem for Stone algebras, Proc. Amer. Math. Soc. , 40 (1973), 75–78. Maddana Swamy, U. and Rama Rao, V. V., Triple and sheaf representations of Stone lattices , Algebra Universalis 5 (1975), 104–113 Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence The retraction Given a residuated lattice A a retraction is a homomorphism h : A ։ S ( A ) onto a subalgebra S ( A ) of A such that h ( h ( a )) = h ( a ) for each a ∈ A . Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence The retraction Given a residuated lattice A a retraction is a homomorphism h : A ։ S ( A ) onto a subalgebra S ( A ) of A such that h ( h ( a )) = h ( a ) for each a ∈ A . If we have a class of residuated lattices with a retraction onto a subalgebra, we have the following situation: Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence h − 1 ( {⊤} ) h S ( A ) A h Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence When can we use S ( A ) and h − 1 ( {⊤} ) to recover A ? Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence When can we use S ( A ) and h − 1 ( {⊤} ) to recover A ? When is it worth using them? Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence When can we use S ( A ) and h − 1 ( {⊤} ) to recover A ? When is it worth using them? Are they all the information we need to characterize A ? Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Boolean skeleton Let A be a bounded residuated lattice. Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Boolean skeleton Let A be a bounded residuated lattice. B ( A ) = { complemented elements of A } Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Boolean skeleton Let A be a bounded residuated lattice. B ( A ) = { complemented elements of A } = { x ∈ A : there exists z ∈ A such that x ∧ z = ⊥ and x ∨ z = ⊤} Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Boolean skeleton Let A be a bounded residuated lattice. B ( A ) = { complemented elements of A } = { x ∈ A : there exists z ∈ A such that x ∧ z = ⊥ and x ∨ z = ⊤} B ( A ) is a subalgebra of A which is a Boolean algebra. Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Stonean residuated lattices Stonean residuated lattices is the greatest subvariety S of bounded residuated lattices that satisfies that for each A ∈ S the application ¬¬ : A → B ( A ) is a retraction onto the boolean skeleton. Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Importance of the Boolean skeleton in Stonean residuated lattices Theorem The following are equivalent conditions for a bounded residuated lattice A : Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Importance of the Boolean skeleton in Stonean residuated lattices Theorem The following are equivalent conditions for a bounded residuated lattice A : A is Stonean, (i) Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Importance of the Boolean skeleton in Stonean residuated lattices Theorem The following are equivalent conditions for a bounded residuated lattice A : A is Stonean, (i) B ( A ) ⊇ ¬ ( A ) := {¬ x : x ∈ A } . (ii) Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Famous Stonean residuated lattices Boolean algebras Pseudocomplemented BL-algebras Product algebras G¨ odel algebras Pseudocomplemented MTL-algebras Stonean Heyting algebras Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Dense elements Let A be in S . Since ¬¬ : A → B ( A ) is a retraction, the kernel, D ( A ) = { x ∈ A : ¬¬ x = ⊤} is a filter of A . Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Dense elements Let A be in S . Since ¬¬ : A → B ( A ) is a retraction, the kernel, D ( A ) = { x ∈ A : ¬¬ x = ⊤} is a filter of A . We will consider D ( A ) = ( D ( A ) , ∗ , → , ∨ , ∧ , ⊤ ) as an integral residuated lattice. Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Building algebras in S Let D be an integral residuated lattice and an element o �∈ D . Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence Building algebras in S Let D be an integral residuated lattice and an element o �∈ D . Adjoining the element o as bottom element, then S ( D ) = ( { o } ∪ D , ∗ , → , ∨ , ∧ , ⊤ , o ) is in S Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence D Manuela Busaniche Blast 2018

Bounded residuated lattices Stonean residuated lattices The general case General categorical equivalence D S ( D ) Manuela Busaniche Blast 2018