Page 69 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 69
II: Skew Lattices
When do these double partitions coincide with the (generally finer) J-M partitions.
Theorem 2.4.10. A skew lattice S is symmetric if and only if for many two
equivalence classes A and B: (i) the double partition of the join class J by intersections of
A-cosets with B-cosets equals the partition by cosets of the meet class M; (ii) the dual assertion
holds for the meet class M.
Proof. First assume that A-cosets and B-cosets in J and M intersect to J-M cosets. Let a ∈A and
b ∈B be given with a∨b = b∨a. By Theorem 2.4.9, a∧b and b∧a lies in the same A-B coset
intersection in M. Thus a∧b and b∧a belong to the same J-coset in M. Since a∨b ≥ both a∧b and
b∧a, we must have a∧b = b∧a. Dually, a∧b = b∧a implies a∨b = b∨a.
Suppose instead that say more than J-coset in M lies inside the intersection I of an
A-coset with a B-coset. Thus there exists j ∈ J with at least two distinct images m and mʹ in I.
We form a subalgebra T = Jʹ ∪ Aʹ ∪ Bʹ ∪ Mʹ inside the subalgebra Sʹ = j∧S∧j as follows. Set Jʹ
= {j}, Mʹ = j∧I∧j and set Aʹ equal to the image of Mʹ in A∩Sʹ under a single coset bijection of Sʹ
from Mʹ to the intermediate class. Define Bʹ in similar fashion. Thus in T exactly one coset
bijection exists from Aʹ to Mʹ and likewise exactly one coset bijection exists from Bʹ to Mʹ. In
particular, exactly one aʹ ∈A and exactly one bʹ ∈B exist such that aʹ ≥ m and bʹ ≥ mʹ. Then
aʹ∨bʹ = bʹ∨aʹ = j and by orthogonality aʹ∧ bʹ = m∧ mʹ and bʹ∧ aʹ = mʹ∧m. Since m ≠ mʹ,
m∧mʹ ≠ mʹ∧m follows and S has an instance of anti-symmetry. £
These results have a number of worthwhile corollaries.
Coset bijections in skew chains and as morphisms in categories
A skew chain is a skew lattice S with finitely many D-classes that is totally quasi-
ordered under ≻ . Thus it can be viewed as a chain of D-classes A > B > C > … > X in that are
totally ordered in S/D. We begin with a near-obvious lemma.
Lemma 2.4.11. Given a skew chain A > B > C:
(1) For each c ∈ C, there is the inclusion of cosets A∧c∧A ⊆ B∧c∧B in C.
(2) For each a ∈ A, there is the inclusion of cosets C∨a∨C ⊆ B∨a∨B in A.
(3) Given a > b > c where a ∈ A, b ∈ B and c ∈ C, if
ϕ: B∨a∨B → A∧b∧A, ψ: C∨b∨C → B∧c∧B and χ: C∨a∨C → A∧c∧A
are coset bijections between the relevant cosets in the respective D-classes taking a to b,
b to c and a to c, then ψoϕ ⊆ χ.
Proof. Given x = a∧c∧a ∈ A∧c∧A, b ∈ B exists such that b∧c∧b = c. Hence, x = a∧b∧c∧b∧a
which us in B∧c∧B and (1) follows. The proof of (2) is dual. To see (3), first observe that the
output set of ϕ and the input set for ψ have the intersection A∧b∧A ∩ C∨b∨C within B. Thus (3)
follows from the inclusions,
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When do these double partitions coincide with the (generally finer) J-M partitions.
Theorem 2.4.10. A skew lattice S is symmetric if and only if for many two
equivalence classes A and B: (i) the double partition of the join class J by intersections of
A-cosets with B-cosets equals the partition by cosets of the meet class M; (ii) the dual assertion
holds for the meet class M.
Proof. First assume that A-cosets and B-cosets in J and M intersect to J-M cosets. Let a ∈A and
b ∈B be given with a∨b = b∨a. By Theorem 2.4.9, a∧b and b∧a lies in the same A-B coset
intersection in M. Thus a∧b and b∧a belong to the same J-coset in M. Since a∨b ≥ both a∧b and
b∧a, we must have a∧b = b∧a. Dually, a∧b = b∧a implies a∨b = b∨a.
Suppose instead that say more than J-coset in M lies inside the intersection I of an
A-coset with a B-coset. Thus there exists j ∈ J with at least two distinct images m and mʹ in I.
We form a subalgebra T = Jʹ ∪ Aʹ ∪ Bʹ ∪ Mʹ inside the subalgebra Sʹ = j∧S∧j as follows. Set Jʹ
= {j}, Mʹ = j∧I∧j and set Aʹ equal to the image of Mʹ in A∩Sʹ under a single coset bijection of Sʹ
from Mʹ to the intermediate class. Define Bʹ in similar fashion. Thus in T exactly one coset
bijection exists from Aʹ to Mʹ and likewise exactly one coset bijection exists from Bʹ to Mʹ. In
particular, exactly one aʹ ∈A and exactly one bʹ ∈B exist such that aʹ ≥ m and bʹ ≥ mʹ. Then
aʹ∨bʹ = bʹ∨aʹ = j and by orthogonality aʹ∧ bʹ = m∧ mʹ and bʹ∧ aʹ = mʹ∧m. Since m ≠ mʹ,
m∧mʹ ≠ mʹ∧m follows and S has an instance of anti-symmetry. £
These results have a number of worthwhile corollaries.
Coset bijections in skew chains and as morphisms in categories
A skew chain is a skew lattice S with finitely many D-classes that is totally quasi-
ordered under ≻ . Thus it can be viewed as a chain of D-classes A > B > C > … > X in that are
totally ordered in S/D. We begin with a near-obvious lemma.
Lemma 2.4.11. Given a skew chain A > B > C:
(1) For each c ∈ C, there is the inclusion of cosets A∧c∧A ⊆ B∧c∧B in C.
(2) For each a ∈ A, there is the inclusion of cosets C∨a∨C ⊆ B∨a∨B in A.
(3) Given a > b > c where a ∈ A, b ∈ B and c ∈ C, if
ϕ: B∨a∨B → A∧b∧A, ψ: C∨b∨C → B∧c∧B and χ: C∨a∨C → A∧c∧A
are coset bijections between the relevant cosets in the respective D-classes taking a to b,
b to c and a to c, then ψoϕ ⊆ χ.
Proof. Given x = a∧c∧a ∈ A∧c∧A, b ∈ B exists such that b∧c∧b = c. Hence, x = a∧b∧c∧b∧a
which us in B∧c∧B and (1) follows. The proof of (2) is dual. To see (3), first observe that the
output set of ϕ and the input set for ψ have the intersection A∧b∧A ∩ C∨b∨C within B. Thus (3)
follows from the inclusions,
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