Page 202 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
skew chain that is not distributive since its homomorphic image U2 is not distributive. The
twisted outcomes for (6) and (8) prevent it from containing copies of any of the Un for n ≥ 2.
5.7 Counting theorems and cancellative skew lattices
In this chapter we have thus far revisited properties that skew lattices might possess. We
began by further studying symmetric skew lattices and then introducing strongly symmetric skew
lattices. We then gave Karin Cvetko-Vah’s proof of Spinks’ Theorem on the equivalence of the
distributive identities (5.2.1) and (5.2.2) for symmetric skew lattices. Symmetry played an
important role in our analysis of cancellation in skew lattices. We saw that simply cancellative
skew diamonds {J > A. B > M} are characterized by a naturally defined bijection
ξ: Comm2(A, B) → ω(J, M),
given by the function ξ(a, b) = (a∨b, a∧b), between the set Comm2(A, B) of all pairs (a, b) in
A×B for which both a∨b = b∨a and a∧b = b∧a, and the set ω(J, M) of all pairs (j, m) in J×M for
which j > m. This is, of course, an instance of the type of counting theorem, where two related
but distinct sets necessarily have the same size. Other counting theorems exist for (fully)
cancellative – and hence symmetric – skew lattices. But first we recall some definitions:
Given a primitive skew lattice A > B. Recall that for any b ∈B, its image set in A is the
set b∨A∨b = {b∨a∨b⎮a ∈ A} which also coincides with {a ∈ A⎮a ≥ b}. Dually, for any a ∈ A,
its image set in B is a∧B∧a = {a∧b∧a⎮b ∈ B} which also coincides with {b ∈ B⎪a ≥ b}. Both
image sets are rectangular subalgebras. Recall from Section 2.4 that:
i) Image sets of all elements from one D-class have the same size in its opposite D-class.
E.g., given b, bʹ in B, |b∨A∨b| = |bʹ∨A∨bʹ|.
ii) Image sets naturally parameterize the cosets of either class in the other. Thus b∨A∨b is a
cross-section of all B-cosets in A while a∧B∧a is a cross-section of all A-cosets in B.
The index of B in A, denoted [A: B], is the size |b∨A∨b| that counts the number of B-cosets in A.
Likewise, the index of A in B, denoted [B: A], is the size |a∧B∧a| that counts counts the number
of A-cosets in B. In general, no direct relationship need exist between [A: B] and [B: A]. The
common size of all A-cosets in B and all B-cosets in A is denoted by ωAB, or equivalently, ωBA.
Clearly the following basic equalities hold:
|A| = [A: B]ωAB and |B| = [B: A]ωAB. (5.7.1)
Given a skew diamond {J > A, B > M} one has five pairs of indices. If the skew diamond
is cancellative, one has the following “Index Laws” that connect opposite pairs of indices. They
are from Karin Cvetko-Vah’s Dissertation [Ref.]. (See also CKLS.)
Theorem 5.7.1. Let S be a cancellative skew diamond {J > A, B > M}. Then:
200
skew chain that is not distributive since its homomorphic image U2 is not distributive. The
twisted outcomes for (6) and (8) prevent it from containing copies of any of the Un for n ≥ 2.
5.7 Counting theorems and cancellative skew lattices
In this chapter we have thus far revisited properties that skew lattices might possess. We
began by further studying symmetric skew lattices and then introducing strongly symmetric skew
lattices. We then gave Karin Cvetko-Vah’s proof of Spinks’ Theorem on the equivalence of the
distributive identities (5.2.1) and (5.2.2) for symmetric skew lattices. Symmetry played an
important role in our analysis of cancellation in skew lattices. We saw that simply cancellative
skew diamonds {J > A. B > M} are characterized by a naturally defined bijection
ξ: Comm2(A, B) → ω(J, M),
given by the function ξ(a, b) = (a∨b, a∧b), between the set Comm2(A, B) of all pairs (a, b) in
A×B for which both a∨b = b∨a and a∧b = b∧a, and the set ω(J, M) of all pairs (j, m) in J×M for
which j > m. This is, of course, an instance of the type of counting theorem, where two related
but distinct sets necessarily have the same size. Other counting theorems exist for (fully)
cancellative – and hence symmetric – skew lattices. But first we recall some definitions:
Given a primitive skew lattice A > B. Recall that for any b ∈B, its image set in A is the
set b∨A∨b = {b∨a∨b⎮a ∈ A} which also coincides with {a ∈ A⎮a ≥ b}. Dually, for any a ∈ A,
its image set in B is a∧B∧a = {a∧b∧a⎮b ∈ B} which also coincides with {b ∈ B⎪a ≥ b}. Both
image sets are rectangular subalgebras. Recall from Section 2.4 that:
i) Image sets of all elements from one D-class have the same size in its opposite D-class.
E.g., given b, bʹ in B, |b∨A∨b| = |bʹ∨A∨bʹ|.
ii) Image sets naturally parameterize the cosets of either class in the other. Thus b∨A∨b is a
cross-section of all B-cosets in A while a∧B∧a is a cross-section of all A-cosets in B.
The index of B in A, denoted [A: B], is the size |b∨A∨b| that counts the number of B-cosets in A.
Likewise, the index of A in B, denoted [B: A], is the size |a∧B∧a| that counts counts the number
of A-cosets in B. In general, no direct relationship need exist between [A: B] and [B: A]. The
common size of all A-cosets in B and all B-cosets in A is denoted by ωAB, or equivalently, ωBA.
Clearly the following basic equalities hold:
|A| = [A: B]ωAB and |B| = [B: A]ωAB. (5.7.1)
Given a skew diamond {J > A, B > M} one has five pairs of indices. If the skew diamond
is cancellative, one has the following “Index Laws” that connect opposite pairs of indices. They
are from Karin Cvetko-Vah’s Dissertation [Ref.]. (See also CKLS.)
Theorem 5.7.1. Let S be a cancellative skew diamond {J > A, B > M}. Then:
200
