Priestley duality for MV-algebras: A new perspective Wesley Fussner (joint work with Mai Gehrke, Sam van Gool, and Vincenzo Marra) CNRS, France Shanks Workshop on Ordered Algebras and Logic Nashville, TN, USA 7 March 2020 1 / 36 DRAFT 2 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. Outline 2 / 36 DRAFT 2 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. Outline 1 Priestley duality as usually presented. 2 / 36 DRAFT 2 3 Priestley duality via the canonical extension. 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. Outline 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 2 / 36 DRAFT 2 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. Outline 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 2 / 36 DRAFT 2 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. Outline 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 4 -algebras. 2 / 36 DRAFT 2 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. Outline 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 2 / 36 DRAFT 2 7 Specializing this to MV-algebras. Outline 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 2 / 36 DRAFT 2 Outline 1 Priestley duality as usually presented. 2 Canonical extensions of bounded distributive lattices. 3 Priestley duality via the canonical extension. 4 -algebras. 5 Priestley duality for -algebras by their canonical extensions. 6 Splitting of the operation on duals to obtain a more expressive environment. 7 Specializing this to MV-algebras. 2 / 36 DRAFT 2 Part I: Priestley duality and canonical extensions 3 / 36 DRAFT 2 Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: Priestley spaces 4 / 36 DRAFT 2 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 4 / 36 DRAFT 2 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 4 / 36 DRAFT 2 Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. 4 / 36 DRAFT 2 If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: 4 / 36 DRAFT 2 The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion. 4 / 36 DRAFT 2 , where a^ = fx 2 X (L): a 2 xg: Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg 4 / 36 DRAFT 2 Priestley spaces Definition: A Priestley space is an ordered topological space (X ; ≤; τ) such that: 1 (X ; τ) is compact, and 2 for every x; y 2 X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 62 U, y 2 U. Generic example: If L = (L; ^; _; 0; 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis fa^; (^a)c : a 2 Lg, where a^ = fx 2 X (L): a 2 xg: 4 / 36 DRAFT 2 The correspondence L 7! X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps). Maps h: L1 ! L2 are sent to continuous isotone maps −1 X (L2) ! X (L1) given by h 7! h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products: R(x; y; z) () x ⊆ #fa · b : a 2 y; b 2 zg: Priestley duality 5 / 36 DRAFT 2 Maps h: L1 ! L2 are sent to continuous isotone maps −1 X (L2) ! X (L1) given by h 7! h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products: R(x; y; z) () x ⊆ #fa · b : a 2 y; b 2 zg: Priestley duality The correspondence L 7! X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps). 5 / 36 DRAFT 2 The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products: R(x; y; z) () x ⊆ #fa · b : a 2 y; b 2 zg: Priestley duality The correspondence L 7! X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).