The structure of skew distributive lattices
Transcript of 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.
Karin Cvetko-Vah The structure of skew distributive lattices
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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.
Karin Cvetko-Vah The structure of skew distributive lattices
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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.
Karin Cvetko-Vah The structure of skew distributive lattices
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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.
Karin Cvetko-Vah The structure of skew distributive lattices
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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
uuuuuu
uuuuu
DDDD
DDDD
x ∧ y ∧ x
IIII
IIII
IIx \ y
{{{{{{{{
0
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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
uuuuuu
uuuuu
DDDD
DDDD
x ∧ y ∧ x
IIII
IIII
IIx \ y
{{{{{{{{
0
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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
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 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.
Karin Cvetko-Vah The structure of skew distributive lattices
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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
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
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 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
��������
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 )..
.. ..
.
.
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]).
Karin Cvetko-Vah The structure of skew distributive lattices