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Esakia Duality for Sugihara Monoids

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Esakia Duality for Sugihara Monoids Esakia duality for Sugihara monoids Wesley Fussner (Joint work with Nick Galatos) University of Denver Department of Mathematics Fall AMS Special Session on Algebraic Logic University of Denver October 8, 2016 1 / 20 DRAFT 2 Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. Kripke semantics is underwritten by Esakia duality for Heyting algebras Motivating Question: Can we construct an Esakia-style duality that lifts Dunn's semantics to a categorical equivalence? Motivation Why another duality for relevant algebras? 2 / 20 DRAFT 2 Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. Kripke semantics is underwritten by Esakia duality for Heyting algebras Motivating Question: Can we construct an Esakia-style duality that lifts Dunn's semantics to a categorical equivalence? Motivation Why another duality for relevant algebras? Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. 2 / 20 DRAFT 2 Kripke semantics is underwritten by Esakia duality for Heyting algebras Motivating Question: Can we construct an Esakia-style duality that lifts Dunn's semantics to a categorical equivalence? Motivation Why another duality for relevant algebras? Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. 2 / 20 DRAFT 2 Motivating Question: Can we construct an Esakia-style duality that lifts Dunn's semantics to a categorical equivalence? Motivation Why another duality for relevant algebras? Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. Kripke semantics is underwritten by Esakia duality for Heyting algebras 2 / 20 DRAFT 2 Motivation Why another duality for relevant algebras? Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. Kripke semantics is underwritten by Esakia duality for Heyting algebras Motivating Question: Can we construct an Esakia-style duality that lifts Dunn's semantics to a categorical equivalence? 2 / 20 DRAFT 2 A Priestley space is called an Esakia space if whenever U is clopen, the down-set #U is also clopen. An Esakia function is a map f : X ! Y between Esakia spaces that is continuous, order-preserving, and satisfies "f (x) ⊆ f ["x] for every x 2 X . Priestley and Esakia duality Definitions: An ordered topological space (X ; ≤; T ) is called a Priestley space if (X ; T ) is compact, and whenever x; y 2 X with x 6≤ y there exists a clopen up-set U ⊆ X with x 2 U, y 2= U. 3 / 20 DRAFT 2 An Esakia function is a map f : X ! Y between Esakia spaces that is continuous, order-preserving, and satisfies "f (x) ⊆ f ["x] for every x 2 X . Priestley and Esakia duality Definitions: An ordered topological space (X ; ≤; T ) is called a Priestley space if (X ; T ) is compact, and whenever x; y 2 X with x 6≤ y there exists a clopen up-set U ⊆ X with x 2 U, y 2= U. A Priestley space is called an Esakia space if whenever U is clopen, the down-set #U is also clopen. 3 / 20 DRAFT 2 Priestley and Esakia duality Definitions: An ordered topological space (X ; ≤; T ) is called a Priestley space if (X ; T ) is compact, and whenever x; y 2 X with x 6≤ y there exists a clopen up-set U ⊆ X with x 2 U, y 2= U. A Priestley space is called an Esakia space if whenever U is clopen, the down-set #U is also clopen. An Esakia function is a map f : X ! Y between Esakia spaces that is continuous, order-preserving, and satisfies "f (x) ⊆ f ["x] for every x 2 X . 3 / 20 DRAFT 2 Theorem (Esakia): The category of Heyting algebras (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Esakia spaces with morphisms the Esakia functions. The Esakia duality is a restricted form of the Priestley duality. Priestley and Esakia duality Theorem (Priestley): The category of bounded distributive lattices (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Priestley spaces with morphisms the continuous, isotone maps. 4 / 20 DRAFT 2 The Esakia duality is a restricted form of the Priestley duality. Priestley and Esakia duality Theorem (Priestley): The category of bounded distributive lattices (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Priestley spaces with morphisms the continuous, isotone maps. Theorem (Esakia): The category of Heyting algebras (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Esakia spaces with morphisms the Esakia functions. 4 / 20 DRAFT 2 Priestley and Esakia duality Theorem (Priestley): The category of bounded distributive lattices (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Priestley spaces with morphisms the continuous, isotone maps. Theorem (Esakia): The category of Heyting algebras (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Esakia spaces with morphisms the Esakia functions. The Esakia duality is a restricted form of the Priestley duality. 4 / 20 DRAFT 2 (A; ·; t) is a commutative monoid, and for all a; b; c 2 A, a · b ≤ c () a ≤ b ! c (A; ^; _) is a lattice, Sugihara monoids Definition: A commutative residuated lattice (CRL) is an algebra (A; ^; _; ·; !; t) such that 5 / 20 DRAFT 2 for all a; b; c 2 A, a · b ≤ c () a ≤ b ! c (A; ·; t) is a commutative monoid, and Sugihara monoids Definition: A commutative residuated lattice (CRL) is an algebra (A; ^; _; ·; !; t) such that (A; ^; _) is a lattice, 5 / 20 DRAFT 2 for all a; b; c 2 A, a · b ≤ c () a ≤ b ! c Sugihara monoids Definition: A commutative residuated lattice (CRL) is an algebra (A; ^; _; ·; !; t) such that (A; ^; _) is a lattice, (A; ·; t) is a commutative monoid, and 5 / 20 DRAFT 2 Sugihara monoids Definition: A commutative residuated lattice (CRL) is an algebra (A; ^; _; ·; !; t) such that (A; ^; _) is a lattice, (A; ·; t) is a commutative monoid, and for all a; b; c 2 A, a · b ≤ c () a ≤ b ! c 5 / 20 DRAFT 2 distributive if its lattice reduct is distributive, idempotent if it satisfies the identity a2 = a. integral if t is the greatest element with respect to the lattice order, Definition: The expansion of a CRL by a unary operation : satisfying ::a = a and a !:b = b !:a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid. For simplicity, we consider expansions of Sugihara monoids by universal bounds ? and >. Sugihara monoids Definition: A CRL is called 6 / 20 DRAFT 2 idempotent if it satisfies the identity a2 = a. distributive if its lattice reduct is distributive, Definition: The expansion of a CRL by a unary operation : satisfying ::a = a and a !:b = b !:a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid. For simplicity, we consider expansions of Sugihara monoids by universal bounds ? and >. Sugihara monoids Definition: A CRL is called integral if t is the greatest element with respect to the lattice order, 6 / 20 DRAFT 2 idempotent if it satisfies the identity a2 = a. Definition: The expansion of a CRL by a unary operation : satisfying ::a = a and a !:b = b !:a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid. For simplicity, we consider expansions of Sugihara monoids by universal bounds ? and >. Sugihara monoids Definition: A CRL is called integral if t is the greatest element with respect to the lattice order, distributive if its lattice reduct is distributive, 6 / 20 DRAFT 2 Definition: The expansion of a CRL by a unary operation : satisfying ::a = a and a !:b = b !:a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid. For simplicity, we consider expansions of Sugihara monoids by universal bounds ? and >. Sugihara monoids Definition: A CRL is called integral if t is the greatest element with respect to the lattice order, distributive if its lattice reduct is distributive, idempotent if it satisfies the identity a2 = a. 6 / 20 DRAFT 2 For simplicity, we consider expansions of Sugihara monoids by universal bounds ? and >. Sugihara monoids Definition: A CRL is called integral if t is the greatest element with respect to the lattice order, distributive if its lattice reduct is distributive, idempotent if it satisfies the identity a2 = a. Definition: The expansion of a CRL by a unary operation : satisfying ::a = a and a !:b = b !:a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid. 6 / 20 DRAFT 2 Sugihara monoids Definition: A CRL is called integral if t is the greatest element with respect to the lattice order, distributive if its lattice reduct is distributive, idempotent if it satisfies the identity a2 = a. Definition: The expansion of a CRL by a unary operation : satisfying ::a = a and a !:b = b !:a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid. For simplicity, we consider expansions of Sugihara monoids by universal bounds ? and >. 6 / 20 DRAFT 2 : satisfies the identities ::a = a, :? = >, :(a ^ b) = :a _:b, a ^ :a ≤ b _:b. (A; ^; _; ?; >) is a bounded distributive lattice, and Proposition: The (^; _; :; ?; >)-reduct of every bounded Sugihara monoid is a Kleene algebra.
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