The structure of skew distributive . lattices .. Karin Cvetko-Vah University of Ljubljana Duality Theory in Algebra, Logic and CS Oxford, August 2012 . Karin Cvetko-Vah The structure of skew distributive lattices . Motivation Skew lattices: noncommutative lattices. Andrej Bauer, KCV, Mai Gehrke, Sam Van Gool and Ganna Kudryavtseva: A non-commutative Priestley duality, preprint. Sam will explain the contents of the paper. Karin Cvetko-Vah The structure of skew distributive lattices . Skew lattices A skew lattice is an algebra (S; ^; _) of type (2; 2) such that ^ and _ are both idempotent and associative, and they dualize each other in that x ^ y = x iff x _ y = y and x ^ y = y iff x _ y = x: A skew lattice with zero is an algebra (S; ^; _; 0) of type (2; 2; 0) such that (S; ^; _) is a skew lattice and x ^ 0 = 0 = 0 ^ x for all x 2 S. Karin Cvetko-Vah The structure of skew distributive lattices . Skew lattices A skew lattice is an algebra (S; ^; _) of type (2; 2) such that ^ and _ are both idempotent and associative, and they dualize each other in that x ^ y = x iff x _ y = y and x ^ y = y iff x _ y = x: A skew lattice with zero is an algebra (S; ^; _; 0) of type (2; 2; 0) such that (S; ^; _) is a skew lattice and x ^ 0 = 0 = 0 ^ x for all x 2 S. Karin Cvetko-Vah The structure of skew distributive lattices . Rectangular bands A rectangular band (A; ^): ^ is idempotent and associative x ^ y ^ z = x ^ z It becomes a skew lattice if we define x _ y = y ^ x. For sets X and Y define ^ on X × Y : (x1; y1) ^ (x2; y2) = (x1; y2) . .u ^ v .v .u .u _ v . (X × Y ; ^) is a rectangular band. Every rectangular band is isomorphic to one such. Karin Cvetko-Vah The structure of skew distributive lattices . Rectangular bands A rectangular band (A; ^): ^ is idempotent and associative x ^ y ^ z = x ^ z It becomes a skew lattice if we define x _ y = y ^ x. For sets X and Y define ^ on X × Y : (x1; y1) ^ (x2; y2) = (x1; y2) . .u ^ v .v .u .u _ v . (X × Y ; ^) is a rectangular band. Every rectangular band is isomorphic to one such. Karin Cvetko-Vah The structure of skew distributive lattices D . The order and Green’s relation S a skew lattice. Natural preorder: x ≼ y iff x ^ y ^ x = x (and dually y _ x _ y = y): Natural partial order: x ≤ y iff x ^ y = x = y ^ x (and dually y _ x = y = x _ y): Green’s relation D is defined by xDy , (x ≼ y and y ≼ x): . Karin Cvetko-Vah The structure of skew distributive lattices D . The order and Green’s relation S a skew lattice. Natural preorder: x ≼ y iff x ^ y ^ x = x (and dually y _ x _ y = y): Natural partial order: x ≤ y iff x ^ y = x = y ^ x (and dually y _ x = y = x _ y): Green’s relation D is defined by xDy , (x ≼ y and y ≼ x): . Karin Cvetko-Vah The structure of skew distributive lattices Leech’s decomposition theorems . The first decomposition theorem . Theorem (Leech, 1989) . .. D is a congruence; S=D is a lattice: the maximal lattice image of S, and each D-class is a rectangular band. .. So: a skew lattice is a lattice of rectangular bands. Fact: x ≼ y in S iff Dx ≤ Dy in S=D. Karin Cvetko-Vah The structure of skew distributive lattices Leech’s decomposition theorems . The first decomposition theorem . Theorem (Leech, 1989) . .. D is a congruence; S=D is a lattice: the maximal lattice image of S, and each D-class is a rectangular band. .. So: a skew lattice is a lattice of rectangular bands. Fact: x ≼ y in S iff Dx ≤ Dy in S=D. Karin Cvetko-Vah The structure of skew distributive lattices Leech’s decomposition theorems . The second decomposition theorem A SL S is: left handed if it satisfies x ^ y ^ x = x ^ y and x _ y _ x = y _ x. right handed if it satisfies x ^ y ^ x = y ^ x and x _ y _ x = x _ y. Most natural examples of SLs are either left or right handed. Leech, 1989: Any SL factors as a fiber product of a left handed SL by a right handed SL over their common maximal lattice image. (Pullback.) . Karin Cvetko-Vah The structure of skew distributive lattices Leech’s decomposition theorems . The second decomposition theorem A SL S is: left handed if it satisfies x ^ y ^ x = x ^ y and x _ y _ x = y _ x. right handed if it satisfies x ^ y ^ x = y ^ x and x _ y _ x = x _ y. Most natural examples of SLs are either left or right handed. Leech, 1989: Any SL factors as a fiber product of a left handed SL by a right handed SL over their common maximal lattice image. (Pullback.) . Karin Cvetko-Vah The structure of skew distributive lattices . Skew distributive lattices A skew distributive lattice is a skew lattice S which satisfies the identities x ^ (y _ z) = (x ^ y) _ (x ^ z); (x _ y) ^ z = (x ^ z) _ (y ^ z): S a skew distributive lattice ) S=D is a distributive lattice. Given any x 2 S: x ^ S ^ x = fx ^ y ^ x j y 2 Sg is a distributive lattice. Karin Cvetko-Vah The structure of skew distributive lattices . Skew distributive lattices A skew distributive lattice is a skew lattice S which satisfies the identities x ^ (y _ z) = (x ^ y) _ (x ^ z); (x _ y) ^ z = (x ^ z) _ (y ^ z): S a skew distributive lattice ) S=D is a distributive lattice. Given any x 2 S: x ^ S ^ x = fx ^ y ^ x j y 2 Sg is a distributive lattice. Karin Cvetko-Vah The structure of skew distributive lattices . Skew Boolean algebras A skew Boolean algebra is an algebra (S; ^; _; n; 0) of type (2; 2; 2; 0) where (S; ^; _; 0) is a skew distributive lattice with 0, x ^ S ^ x = fx ^ y ^ x j y 2 Sg is a Boolean lattice for all x, and x n y is the complement of x ^ y ^ x in x ^ S ^ x. u x D uu DD uu D uu DD uu DD uu x ^ y ^ x x n y II II {{ II {{ II {{ II {{ 0 . Karin Cvetko-Vah The structure of skew distributive lattices . Skew Boolean algebras A skew Boolean algebra is an algebra (S; ^; _; n; 0) of type (2; 2; 2; 0) where (S; ^; _; 0) is a skew distributive lattice with 0, x ^ S ^ x = fx ^ y ^ x j y 2 Sg is a Boolean lattice for all x, and x n y is the complement of x ^ y ^ x in x ^ S ^ x. u x D uu DD uu D uu DD uu DD uu x ^ y ^ x x n y II II {{ II {{ II {{ II {{ 0 . Karin Cvetko-Vah The structure of skew distributive lattices . Skew Boolean algebras A skew Boolean algebra is an algebra (S; ^; _; n; 0) of type (2; 2; 2; 0) where (S; ^; _; 0) is a skew distributive lattice with 0, x ^ S ^ x = fx ^ y ^ x j y 2 Sg is a Boolean lattice for all x, and x n y is the complement of x ^ y ^ x in x ^ S ^ x. u x D uu DD uu D uu DD uu DD uu x ^ y ^ x x n y II II {{ II {{ II {{ II {{ 0 . Karin Cvetko-Vah The structure of skew distributive lattices . Example of a skew Boolean algebra: X * Y For partial maps f ; g : X * Y define: 0 = ; ^ f g = f domf \domg _ [ f g = f domgndomf g n f g = f domf ndomg With these operations X * Y is a skew Boolean algebra. The maximal lattice image: (X * Y )=D = P(X). X * Y is left-handed: f ^ g ^ f = f ^ g. Karin Cvetko-Vah The structure of skew distributive lattices . Duality for skew Boolean algebras A. Bauer, KCV, G. Kudryavtseva: A Boolean space is a locally compact zero-dimensional Hausdorff space. A Boolean sheaf is a local homeomorphism p : E ! B where B is a Boolean space. We only consider sheaves for which p : E ! B is surjective. Theorem: Boolean sheaves are dual to left-handed skew Boolean algebras. Karin Cvetko-Vah The structure of skew distributive lattices . Duality for skew Boolean algebras A. Bauer, KCV, G. Kudryavtseva: A Boolean space is a locally compact zero-dimensional Hausdorff space. A Boolean sheaf is a local homeomorphism p : E ! B where B is a Boolean space. We only consider sheaves for which p : E ! B is surjective. Theorem: Boolean sheaves are dual to left-handed skew Boolean algebras. Karin Cvetko-Vah The structure of skew distributive lattices . Duality for skew Boolean algebras A. Bauer, KCV, G. Kudryavtseva: A Boolean space is a locally compact zero-dimensional Hausdorff space. A Boolean sheaf is a local homeomorphism p : E ! B where B is a Boolean space. We only consider sheaves for which p : E ! B is surjective. Theorem: Boolean sheaves are dual to left-handed skew Boolean algebras. Karin Cvetko-Vah The structure of skew distributive lattices . Duality for finite skew Boolean algebras S a finite left-handed SBA, p : E ! B its dual Boolean sheaf. The points of B are the atoms of S=D. Given an atom D-class A in S: the stalk above A consists of the elements of A.