Page 167 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 167
V: Further Topics in Skew Lattices
lattices are studied in the last part of the Section and Theorem 5.4.7 gives a number of equivalent
conditions for a skew lattice to belong to this variety. Perhaps the most notable one is the unique
midpoint condition on skew chains A > B > C in S: given a > c with a ∈A and c ∈C, a unique
midpoint b ∈B exists such that a > b > c. This is equivalent to the condition: for each pair e ≤ f in
a skew lattice, the interval subalgebra [e, f] ={x ∈S⎪e ≤ x ≤ f} is a sublattice. As a consequence,
any strictly categorical skew lattice S for which the lattice image S/D is distributive is a
distributive skew lattice. (Theorem 5.4.9.) All normal skew lattices belong to this variety as do
their duals, conormal skew lattices (where x∨y∨z∨w = x∨z∨y∨w holds). The subvariety of all
skew lattices generated from these two classes of skew lattices turns out to be a proper subvariety
of strictly categorical skew lattices. At the end of the section we ask if this generated subvariety
is the variety of paranormal skew lattices (defined there). In general one has the following chain
of subvarieties:
〈Normal ∪ conormal〉 ⊆ Paranormal ⊂ Strictly categorical.
In Section 5.5 we take a closer look at distributivity. Every distributive skew lattice S
must be quasi-distributive in that every lattice image of S is distributive. Quasi-distributivity is a
necessary, but not sufficient condition for distributivity, thanks to Spinks’ examples (Theorem
1.3.10). A possible complementary condition is for a skew lattice to be linearly distributive in
that each subalgebra that is totally pre-ordered (by ≻) is distributive. Linearly distributive skew
lattices form a variety. (See Theorems 5.5.1 and 5.5.5.) For skew lattices one has the strict chain
of subvarieties:
Strictly categorical ⊂ Linearly distributive ⊂ Categorical.
In the strictly categorical case, quasi-distributivity suffices for the skew lattice to be distributive
(Theorem 5.4.9). For linearly distributive algebras in general this does not occur (Spinks’
examples again). However: given symmetry, linearly distributive + quasi-distributive implies
distributive. (See Theorem 5.5.11.)
Being a vital part of distributivity, linear distributivity and distributive skew chains in
particular are studied further in Section 5.6. Given a skew chain A > B > C with a > c for a ∈ A
and c ∈ C, consider the set µ(a, c) = {b ∈ B⎪a > b > c} of midpoints b of a and c in B. One has
|µ(a, c)| ≥ 1, and unless A > B > C is strictly categorical, |µ(a, c)| ≥ 2 for all such pairs. In
general, B decomposes into a disjoint union of AC-components B1, B2, … that induce sub-skew
chains A > Bi > C. (The definitions and relevant discussion occur prior to Example 5.4.3.) Each
component Bi contains at least one midpoint bi for a > c. In any case, A > B > C is distributive if
and only if each A > Bi > C is strictly categorical, and in particular, each component Bi contains a
unique midpoint bi for each such a > c. µ(a, c) thus parameterizes in a natural way the AC-
components of B. In so doing, distributivity minimizes the number of midpoints any a > c can
have in a skew chain A > B > C relative to the AC-component structure of B (Theorem 5.6.4).
Here, and in the latter part of Section 5.4, the orthogonality of cosets from two D-classes (A and
C) in a third D-class (B) occurs again. (Recall Lemma 2.4.8 and Theorem 2.4.9 in Section 2.4.)
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lattices are studied in the last part of the Section and Theorem 5.4.7 gives a number of equivalent
conditions for a skew lattice to belong to this variety. Perhaps the most notable one is the unique
midpoint condition on skew chains A > B > C in S: given a > c with a ∈A and c ∈C, a unique
midpoint b ∈B exists such that a > b > c. This is equivalent to the condition: for each pair e ≤ f in
a skew lattice, the interval subalgebra [e, f] ={x ∈S⎪e ≤ x ≤ f} is a sublattice. As a consequence,
any strictly categorical skew lattice S for which the lattice image S/D is distributive is a
distributive skew lattice. (Theorem 5.4.9.) All normal skew lattices belong to this variety as do
their duals, conormal skew lattices (where x∨y∨z∨w = x∨z∨y∨w holds). The subvariety of all
skew lattices generated from these two classes of skew lattices turns out to be a proper subvariety
of strictly categorical skew lattices. At the end of the section we ask if this generated subvariety
is the variety of paranormal skew lattices (defined there). In general one has the following chain
of subvarieties:
〈Normal ∪ conormal〉 ⊆ Paranormal ⊂ Strictly categorical.
In Section 5.5 we take a closer look at distributivity. Every distributive skew lattice S
must be quasi-distributive in that every lattice image of S is distributive. Quasi-distributivity is a
necessary, but not sufficient condition for distributivity, thanks to Spinks’ examples (Theorem
1.3.10). A possible complementary condition is for a skew lattice to be linearly distributive in
that each subalgebra that is totally pre-ordered (by ≻) is distributive. Linearly distributive skew
lattices form a variety. (See Theorems 5.5.1 and 5.5.5.) For skew lattices one has the strict chain
of subvarieties:
Strictly categorical ⊂ Linearly distributive ⊂ Categorical.
In the strictly categorical case, quasi-distributivity suffices for the skew lattice to be distributive
(Theorem 5.4.9). For linearly distributive algebras in general this does not occur (Spinks’
examples again). However: given symmetry, linearly distributive + quasi-distributive implies
distributive. (See Theorem 5.5.11.)
Being a vital part of distributivity, linear distributivity and distributive skew chains in
particular are studied further in Section 5.6. Given a skew chain A > B > C with a > c for a ∈ A
and c ∈ C, consider the set µ(a, c) = {b ∈ B⎪a > b > c} of midpoints b of a and c in B. One has
|µ(a, c)| ≥ 1, and unless A > B > C is strictly categorical, |µ(a, c)| ≥ 2 for all such pairs. In
general, B decomposes into a disjoint union of AC-components B1, B2, … that induce sub-skew
chains A > Bi > C. (The definitions and relevant discussion occur prior to Example 5.4.3.) Each
component Bi contains at least one midpoint bi for a > c. In any case, A > B > C is distributive if
and only if each A > Bi > C is strictly categorical, and in particular, each component Bi contains a
unique midpoint bi for each such a > c. µ(a, c) thus parameterizes in a natural way the AC-
components of B. In so doing, distributivity minimizes the number of midpoints any a > c can
have in a skew chain A > B > C relative to the AC-component structure of B (Theorem 5.6.4).
Here, and in the latter part of Section 5.4, the orthogonality of cosets from two D-classes (A and
C) in a third D-class (B) occurs again. (Recall Lemma 2.4.8 and Theorem 2.4.9 in Section 2.4.)
165
