Page 203 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 203
Further Topics in Skew Lattices
[M: A] = [B: J] and [A: M] = [J: B]. (5.7.2)
[M: B] = [A: J] and [B: M] = [J: A]. (5.7.3)
In detail, given j ∈ J, a ∈ A, b ∈ B and m ∈ M such that j > a, b > m, isomorphisms
α: {x ∈ A⎪x > m} ≅ {y ∈ J⎪y > b} and β:{u ∈ B⎪u < j} ≅ {v ∈ M⎪v < a}
are defined by α(x) = x∨b∨x and β(u) = u∧a∧u. Isomorphisms between other pairs of image sets
are defined similarly.
Conversely, if all such maps in a skew diamond are isomorphisms, then it is cancellative.
Proof. Since b > m and S is symmetric, b commutes under both ∧ and ∨ with all x ∈ A such that
x > m so that α is well-defined. Since S is cancellative, α is also one-to-one. Lemma 2.1.4
implies
α(x ∨ xʹ) = (x ∨ xʹ) ∨ b ∨ (x ∨ xʹ) = (x ∨ b ∨ x) ∨ (xʹ ∨ b ∨ xʹ) = α(x) ∨ α(xʹ).
Thus α is a ∨-homomorphism that must be a ∧-homomorphism since index sets are rectangular
subalgebras. To show it is onto and thus an isomorphism, let y > b be given in J and set c =
y∧a∧y in A. Then m < c < y and y = α(c) by Theorem 2.2.1. The verification that β is an
isomorphism is similar.
Conversely, given a skew diamond, if all such maps are at least bijections, copies of
NCR5 , NCL5 and the algebras NSR7 ,0 , NSR7 ,1 , NSL7 ,0 and NSL7 ,1 cannot occur as subalgebras.
Since any skew diamond is trivially quasi-distributive, it must be cancellative. £
What else can be said? First, observe:
Theorem 5.7.2. A skew diamond is simply cancellative if and only if it is strictly
categorical, in which case it is also distributive. It is cancellative if and only if it is symmetric
and strictly categorical.
Proof. A skew diamond is already quasi-distributive. To be simply cancellative it needs to
exclude both NC5 subalgebras, and to be strictly categorical it needs to exclude both 4-element
skew chains in Theorem 5.4.8(iv). But both constraints are equivalent for skew diamonds.
Theorem 5.4.10 insures the addendum of distributivity. £
The following two results from João Pita da Costa’s dissertation are relevant.
Theorem 5.7.3 Given a strictly categorical skew chain A > B > C, if A and C are finite,
then so is B and
|B| = ωABωBC = [B : A] ωAC [B : C].
ω AC
In general, a strictly categorical skew lattice S is finite if and only if its maximal lattice image
S/D is finite and both the maximal and minimal D-classes in S are finite.
201
[M: A] = [B: J] and [A: M] = [J: B]. (5.7.2)
[M: B] = [A: J] and [B: M] = [J: A]. (5.7.3)
In detail, given j ∈ J, a ∈ A, b ∈ B and m ∈ M such that j > a, b > m, isomorphisms
α: {x ∈ A⎪x > m} ≅ {y ∈ J⎪y > b} and β:{u ∈ B⎪u < j} ≅ {v ∈ M⎪v < a}
are defined by α(x) = x∨b∨x and β(u) = u∧a∧u. Isomorphisms between other pairs of image sets
are defined similarly.
Conversely, if all such maps in a skew diamond are isomorphisms, then it is cancellative.
Proof. Since b > m and S is symmetric, b commutes under both ∧ and ∨ with all x ∈ A such that
x > m so that α is well-defined. Since S is cancellative, α is also one-to-one. Lemma 2.1.4
implies
α(x ∨ xʹ) = (x ∨ xʹ) ∨ b ∨ (x ∨ xʹ) = (x ∨ b ∨ x) ∨ (xʹ ∨ b ∨ xʹ) = α(x) ∨ α(xʹ).
Thus α is a ∨-homomorphism that must be a ∧-homomorphism since index sets are rectangular
subalgebras. To show it is onto and thus an isomorphism, let y > b be given in J and set c =
y∧a∧y in A. Then m < c < y and y = α(c) by Theorem 2.2.1. The verification that β is an
isomorphism is similar.
Conversely, given a skew diamond, if all such maps are at least bijections, copies of
NCR5 , NCL5 and the algebras NSR7 ,0 , NSR7 ,1 , NSL7 ,0 and NSL7 ,1 cannot occur as subalgebras.
Since any skew diamond is trivially quasi-distributive, it must be cancellative. £
What else can be said? First, observe:
Theorem 5.7.2. A skew diamond is simply cancellative if and only if it is strictly
categorical, in which case it is also distributive. It is cancellative if and only if it is symmetric
and strictly categorical.
Proof. A skew diamond is already quasi-distributive. To be simply cancellative it needs to
exclude both NC5 subalgebras, and to be strictly categorical it needs to exclude both 4-element
skew chains in Theorem 5.4.8(iv). But both constraints are equivalent for skew diamonds.
Theorem 5.4.10 insures the addendum of distributivity. £
The following two results from João Pita da Costa’s dissertation are relevant.
Theorem 5.7.3 Given a strictly categorical skew chain A > B > C, if A and C are finite,
then so is B and
|B| = ωABωBC = [B : A] ωAC [B : C].
ω AC
In general, a strictly categorical skew lattice S is finite if and only if its maximal lattice image
S/D is finite and both the maximal and minimal D-classes in S are finite.
201
