The structure of skew distributive lattices

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The structure of skew distributivelattices

Karin Cvetko-Vah

University of Ljubljana

Duality Theory in Algebra, Logic and CSOxford, August 2012

Karin Cvetko-Vah The structure of skew distributive lattices

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.. Motivation

Skew lattices: noncommutative lattices.

Andrej Bauer, KCV, Mai Gehrke, Sam Van Gool andGanna Kudryavtseva: A non-commutative Priestleyduality, preprint.

Sam will explain the contents of the paper.


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.. Skew lattices

A skew lattice is an algebra (S;∧,∨) of type (2,2) suchthat ∧ and ∨ are both idempotent and associative, andthey dualize each other in that

x ∧ y = x iff x ∨ y = y andx ∧ 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 andx ∧ 0 = 0 = 0 ∧ x for all x ∈ S.


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.. Skew lattices

A skew lattice is an algebra (S;∧,∨) of type (2,2) suchthat ∧ and ∨ are both idempotent and associative, andthey dualize each other in that

x ∧ y = x iff x ∨ y = y andx ∧ 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 andx ∧ 0 = 0 = 0 ∧ x for all x ∈ S.


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.. Rectangular bands

A rectangular band (A,∧):∧ is idempotent and associativex ∧ 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

..u ∧ v

..v

..u ∨ v

(X × Y ,∧) is a rectangular band.Every rectangular band is isomorphic to one such.


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.. Rectangular bands

A rectangular band (A,∧):∧ is idempotent and associativex ∧ 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

..u ∧ v

..v

..u ∨ v

(X × Y ,∧) is a rectangular band.Every rectangular band is isomorphic to one such.


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.. The order and Green’s relation D

S a skew lattice.

Natural preorder: x ≼ y iff x ∧ y ∧ x = x (and duallyy ∨ x ∨ y = y).

Natural partial order: x ≤ y iff x ∧ y = x = y ∧ x (anddually y ∨ x = y = x ∨ y).

Green’s relation D is defined by

xDy ⇔ (x ≼ y and y ≼ x).


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.. The order and Green’s relation D

S a skew lattice.

Natural preorder: x ≼ y iff x ∧ y ∧ x = x (and duallyy ∨ x ∨ y = y).

Natural partial order: x ≤ y iff x ∧ y = x = y ∧ x (anddually y ∨ x = y = x ∨ y).

Green’s relation D is defined by

xDy ⇔ (x ≼ y and y ≼ x).


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..Leech’s decomposition theoremsThe first decomposition theorem

.Theorem (Leech, 1989)..

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D is a congruence;S/D is a lattice: the maximal lattice image of S, andeach 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.


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..Leech’s decomposition theoremsThe first decomposition theorem

.Theorem (Leech, 1989)..

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D is a congruence;S/D is a lattice: the maximal lattice image of S, andeach 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.


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..Leech’s decomposition theoremsThe second decomposition theorem

A SL S is:left handed if it satisfies x ∧ y ∧ x = x ∧ y andx ∨ y ∨ x = y ∨ x .right handed if it satisfies x ∧ y ∧ x = y ∧ x andx ∨ y ∨ x = x ∨ y .

Most natural examples of SLs are either left or righthanded.

Leech, 1989: Any SL factors as a fiber product of a lefthanded SL by a right handed SL over their commonmaximal lattice image. (Pullback.)


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..Leech’s decomposition theoremsThe second decomposition theorem

A SL S is:left handed if it satisfies x ∧ y ∧ x = x ∧ y andx ∨ y ∨ x = y ∨ x .right handed if it satisfies x ∧ y ∧ x = y ∧ x andx ∨ y ∨ x = x ∨ y .

Most natural examples of SLs are either left or righthanded.

Leech, 1989: Any SL factors as a fiber product of a lefthanded SL by a right handed SL over their commonmaximal lattice image. (Pullback.)


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.. Skew distributive lattices

A skew distributive lattice is a skew lattice S whichsatisfies 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 adistributive lattice.


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.. Skew distributive lattices

A skew distributive lattice is a skew lattice S whichsatisfies 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 adistributive lattice.


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.. Skew Boolean algebras

A skew Boolean algebra is an algebra (S;∧,∨, \, 0) oftype (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 forall x , andx \ y is the complement of x ∧ y ∧ x in x ∧ S ∧ x .

x

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DDDD

DDDD

x ∧ y ∧ x

IIII

IIII

IIx \ y

{{{{{{{{

0


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x

uuuuuu

uuuuu

DDDD

DDDD

x ∧ y ∧ x

IIII

IIII

IIx \ y

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0


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x

uuuuuu

uuuuu

DDDD

DDDD

x ∧ y ∧ x

IIII

IIII

IIx \ y

{{{{{{{{

0


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.. 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 Booleanalgebra.The maximal lattice image: (X ⇀ Y )/D = P(X ).X ⇀ Y is left-handed: f ∧ g ∧ f = f ∧ g.


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.. Duality for skew Boolean algebras

A. Bauer, KCV, G. Kudryavtseva:

A Boolean space is a locally compactzero-dimensional Hausdorff space.A Boolean sheaf is a local homeomorphismp : E → B where B is a Boolean space.We only consider sheaves for which p : E → B issurjective.Theorem:Boolean sheaves are dual to left-handed skewBoolean algebras.


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.. Duality for finite skew Boolean algebrasS a finite left-handed SBA, p : E → B its dual Booleansheaf.

The points of B are the atoms of S/D.Given an atom D-class A in S: the stalk above Aconsists of the elements of A.The elements of S correspond to sections of p(above clopens).

bO

b b

b b b bb b

b b b

b b

b

b

b

b

b


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.. The skew distributive case

Can we do the same with skew distributive lattices?(Taking join irreducibles instead of atoms.)

bO

b b

b b b bb b

b b

b

b

b

b

b

b

b

b

We get too many sections over clopen downsets!


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.. 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 aunigue b ∈ B s. t. b < a.Given a coset Ai of B in A and b ∈ B there exists aunigue a ∈ Ai s. t. b < a.Given a coset Ai of B in A and b ∈ B the cosetbijection φi : Ai → B is defined by a 7→ a ∧ b ∧ a.φi(a) is the element b ∈ B with the property b ≤ a.


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.. The cosets



B ∨ a ∨ B = {b ∨ a ∨ b′ |b, b′ ∈ B}.



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.. The cosets



B ∨ a ∨ B = {b ∨ a ∨ b′ |b, b′ ∈ B}.



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.. The coset decompositionS left-handed DSL, A > B D-classesOperations on A ∪ B : a ∈ Ai , b ∈ B, φi : Ai → B the cosetbijection.Then:

b ∧ a = b, a ∧ b = φi(a), b ∨ a = a, and a ∨ b = φ−1i (b).

B

A1 A2 A3

b

b

b b


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.. ExampleConsider the left-handed skew distributive lattice S:

A

B

S S/D

0b O

b a b b

bd

be

b f bg

bc

b h b A

b B

b


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.. Example

Assume the following coset decomposition:

A

B

b O

b a b b

bd b e b f bg

bc b h

Take the cosets as the elements of the stalk.


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We obtain:

bO

b b

b b b bb b

b b

b

b b

b

b


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.. Completing to a SBA

Forgetting the order on the base space, the sections yielda SBA, a "Booleanization" of S:

A

B

b O

ba

bb

bd

be

bf

bg

bc

bh

bu

bv b w

u, v and w correspond exactly to cosets of B in A.


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Denote by [A : B] the set of all cosets of B in A.

The "Booleanization":

A

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FFFF

FFFF

F

B

====

====

[A : B]

yyyyyyyyy

0


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.. 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 iffB ∨ a ∨ B = B ∨ a′ ∨ B and C ∨ b ∨ C = C ∨ b′ ∨ C fora > b, a′ > b′.

C

B

A ba ba′

bb bb′

So: a coset of C in A corresponds to a pair of a coset of Cin B and a coset of B in A.


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.. The coset law for a diamondA,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 iffB ∨ (b ∨ a) ∨ B = B ∨ (b ∨ a′) ∨ B (for any b ∈ B).

A B

J

M

bbva bbva′

ba ba′b b


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.. The cosets in the dual spaceTo 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.

A

B

<bO

ba

bb

bd

be

bf

bg

bc

bh

bu

bv

bw

b b

b

b

b

b

b


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.. The Mac Neille Theorem

L a DL, B the Booleanization of L..Theorem (Mac Neille, 1945 )..

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Every element x ∈ B can be expressed asx = a1 + · · ·+ an for some a1 ≤ · · · ≤ an, ai ∈ L.

0

1

a a a’

1

0

b

b

b

b

b

b

b

a′ = a + 1.


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.. Completing a DL to a BA

A2

A1

A0

0

A2

PPPPPP

PPPPPP

P

qqqqqq

qqqqqq

A1

MMMMMM

MMMMM A0 + A1 + A2

qqqqqq

qqqqq

PPPPPP

PPPPPP

A0 + A2

nnnnnn

nnnnnn

A0

MMMMMM

MMMMMM

M A0 + A1 A1 + A2

nnnnnn

nnnnnn

nn

0


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.. Completing a SDL to a SBA

A2

A1

A0

0

A2

MMMMMM

MMMMM

vvvvvvvvvv

A1

HHHH

HHHH

H X

vvvvvvvvvv

MMMMMM

MMMMMM

[A2 : A0]

rrrrrr

rrrr

A0

IIII

IIII

III [A1 : A0] [A2 : A1]

qqqqqq

qqqqqq

0

Here:X = [A2 : [A1 : A0]] = ([A2 : A1], [A1 : [A1 : A0]]) =([A2 : A1], [A0 : 0]).


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The structure of skew distributive lattices