. . 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 ∈ 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 ∈ 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 ∈ S: x ∧ S ∧ x = {x ∧ y ∧ x | y ∈ S} 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 ∈ S: x ∧ S ∧ x = {x ∧ y ∧ x | y ∈ S} is a distributive lattice.
...... Karin Cvetko-Vah The structure of skew distributive lattices . Skew Boolean algebras
A skew Boolean algebra is an algebra (S; ∧, ∨, \, 0) of type (2, 2, 2, 0) where (S; ∧, ∨, 0) is a skew distributive lattice with 0, x ∧ S ∧ x = {x ∧ y ∧ x | y ∈ S} is a Boolean lattice for all x, and x \ 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 \ 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; ∧, ∨, \, 0) of type (2, 2, 2, 0) where (S; ∧, ∨, 0) is a skew distributive lattice with 0, x ∧ S ∧ x = {x ∧ y ∧ x | y ∈ S} is a Boolean lattice for all x, and x \ 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 \ 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; ∧, ∨, \, 0) of type (2, 2, 2, 0) where (S; ∧, ∨, 0) is a skew distributive lattice with 0, x ∧ S ∧ x = {x ∧ y ∧ x | y ∈ S} is a Boolean lattice for all x, and x \ 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 \ 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 domg\domf g \ f g = f domf \domg
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. The elements of S correspond to sections of p (above clopens).
b b b b b b b
b b
b b b b b b b
O b b b
...... Karin Cvetko-Vah The structure of skew distributive lattices . The skew distributive case Can we do the same with skew distributive lattices? (Taking join irreducibles instead of atoms.)
b
b
b b b b b b b
b
b b b b
b b O b
b b
We get too many sections over clopen downsets! ...... Karin Cvetko-Vah The structure of skew distributive lattices . The cosets S SDL with 0; A, B D-classes, A > B in S/D, a ∈ A.
The coset of B in A containing a:
B ∨ a ∨ B = {b ∨ a ∨ b′ | b, b′ ∈ B}.
The cosets of B in A form a partition of A.
Given a coset Ai of B in A and a ∈ Ai there exists a unigue b ∈ B s. t. b < a.
Given a coset Ai of B in A and b ∈ B there exists a unigue a ∈ Ai s. t. b < a.
Given a coset Ai of B in A and b ∈ B the coset bijection φi : Ai → B is defined by a 7→ a ∧ b ∧ a.
φi (a) is the element b ∈ B with the property b ≤ a.
...... Karin Cvetko-Vah The structure of skew distributive lattices . The cosets S SDL with 0; A, B D-classes, A > B in S/D, a ∈ A.
The coset of B in A containing a:
B ∨ a ∨ B = {b ∨ a ∨ b′ | b, b′ ∈ B}.
The cosets of B in A form a partition of A.
Given a coset Ai of B in A and a ∈ Ai there exists a unigue b ∈ B s. t. b < a.
Given a coset Ai of B in A and b ∈ B there exists a unigue a ∈ Ai s. t. b < a.
Given a coset Ai of B in A and b ∈ B the coset bijection φi : Ai → B is defined by a 7→ a ∧ b ∧ a.
φi (a) is the element b ∈ B with the property b ≤ a.
...... Karin Cvetko-Vah The structure of skew distributive lattices . The cosets S SDL with 0; A, B D-classes, A > B in S/D, a ∈ A.
The coset of B in A containing a:
B ∨ a ∨ B = {b ∨ a ∨ b′ | b, b′ ∈ B}.
The cosets of B in A form a partition of A.
Given a coset Ai of B in A and a ∈ Ai there exists a unigue b ∈ B s. t. b < a.
Given a coset Ai of B in A and b ∈ B there exists a unigue a ∈ Ai s. t. b < a.
Given a coset Ai of B in A and b ∈ B the coset bijection φi : Ai → B is defined by a 7→ a ∧ b ∧ a.
φi (a) is the element b ∈ B with the property b ≤ a.
...... Karin Cvetko-Vah The structure of skew distributive lattices . The coset decomposition S left-handed DSL, A > B D-classes Operations on A ∪ B : a ∈ Ai , b ∈ B, φi : Ai → B the coset bijection. Then: ∧ ∧ ∨ ∨ −1 b a = b, a b = φi (a), b a = a, and a b = φi (b).
b b b
A1 A2 A3
b B
...... Karin Cvetko-Vah The structure of skew distributive lattices . Example Consider the left-handed skew distributive lattice S:
S S/D
c d e g b b b f b b b h A A b
a b b b B B b
O b b 0
...... Karin Cvetko-Vah The structure of skew distributive lattices . Example
Assume the following coset decomposition:
c d e g b b b f b h A b b
a b B b b
O b
Take the cosets as the elements of the stalk.
...... Karin Cvetko-Vah The structure of skew distributive lattices We obtain:
b b b b b b
b
b b b b
b b O b
b b
...... Karin Cvetko-Vah The structure of skew distributive lattices . Completing to a SBA Forgetting the order on the base space, the sections yield a SBA, a "Booleanization" of S:
c d e f g b b b h A b b b
b a b b u v w B b b b
O b u, v and w correspond exactly to cosets of B in A.
...... Karin Cvetko-Vah The structure of skew distributive lattices Denote by [A : B] the set of all cosets of B in A.
The "Booleanization":
A FF ¡¡ FF ¡¡ FF ¡¡ FF ¡¡ F B = [A : B] == y = yy == yy == yy yy 0
...... Karin Cvetko-Vah The structure of skew distributive lattices . The coset law for a chain Let A > B > C be a chain of D-classes in a SDL S. a, a′ ∈ A.
Pita Costa, 2011: C ∨ a ∨ C = C ∨ a′ C iff B ∨ a ∨ B = B ∨ a′ ∨ B and C ∨ b ∨ C = C ∨ b′ ∨ C for a > b, a′ > b′.
′
b a b a A
′
b b b b B
C So: a coset of C in A corresponds to a pair of a coset of C in B and a coset of B in A...... Karin Cvetko-Vah The structure of skew distributive lattices . The coset law for a diamond A, B D-classes in a MD LH SL, M = A ∧ B, J = A ∨ B in S/D, a, a′ ∈ A. KCV and Pita Costa, 2011: M ∨ a ∨ M = M ∧ a′ ∨ M iff B ∨ (b ∨ a) ∨ B = B ∨ (b ∨ a′) ∨ B (for any b ∈ B).
′
bvab b bva J
′
b a b a b b A B
M
...... Karin Cvetko-Vah The structure of skew distributive lattices . The cosets in the dual space To obtain the dual of a finite SDL S: . ..1 Take the join irreducible D-classes. . ..2 Take the (n-tuples of) cosets along chains in S.
c d e f g h b b b b b b A b
b b a b b b u v w B b b b b b
O b b b <
...... Karin Cvetko-Vah The structure of skew distributive lattices . The Mac Neille Theorem
L a DL, B the Booleanization of L. . Theorem (Mac Neille, 1945 ) . .. Every element x ∈ B can be expressed as ··· ≤ · · · ≤ ∈ x. = a1 + + an for some a1 an, ai L. .. . .
b 1 b 1
b a a b b a’
b b 0 0 a′ = a + 1.
...... Karin Cvetko-Vah The structure of skew distributive lattices . Completing a DL to a BA
A A P 2 qq 2 PPP qqq PPP qq PPP qq PPP qqq A1 A1 MM A0 + A1 + A2 A0 + A2 MM qq PPP nn MM qq PPP nnn qMqMM nnPnPP qq MM nnn PPP qqq n A0 A M A + A A + A 0 MMM 0 1 nn 1 2 MM nnn MMM nnn MMM nnn M nnn 0 0
...... Karin Cvetko-Vah The structure of skew distributive lattices . Completing a SDL to a SBA
A2 A2 M vv MMM vv MM vv MM vv MM vv M A1 A1 H X MM [A2 : A0] HH vv MM r HH vv MMM rrr vHvH rrMM vv HH rr MM vv rr M A 0 A0 II [A1 : A0] [A2 : A1] II qq II qq II qqq II qq I qqq 0 0 Here: X = [A2 :[A1 : A0]] = ([A2 : A1], [A1 :[A1 : A0]]) = ([A2 : A1], [A0 : 0]).
...... Karin Cvetko-Vah The structure of skew distributive lattices