Page 72 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 72
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 2.4.16. Normal skew lattices are strictly categorical.
Proof. Since the lower D-class in any maximal primitive subalgebra of a normal skew lattice has
exactly one coset, the composition of adjacent coset bijections is a nonempty coset bijection. £
A third class of categorical skew lattices is as follows.
Theorem 2.4.17. Every primitive skew lattice is strictly categorical, distributive and
symmetric. Thus all skew lattices in the subvariety generated from the class of primitive skew
lattices are categorical, distributive and symmetric.
Proof. Any primitive skew lattice S is trivially strictly categorical. The only nontrivial instances
of either x∨y = y∨x or x∧y = y∧x when S is primitive are when x > y or y > x. Thus S is also
symmetric. Next, consider the equation x∧(y∨z)∧x = (x∧y∧x)∨(x∧z∧x). It holds trivially when x
is in the lower D-class B. So let x = a in the upper class A and let y = b and z = c in B. Then
a∧(b∨c)∧a = a∧(c∧b)∧a = (a∧c∧a) ∧ (a∧b∧a) = (a∧b∧a) ∨ (a∧c∧a) in B.
If a in A and say b in B but c in A then b∨c in A so that a∧(b∨c)∧a = a, while (a∧b∧a) ∨ (a∧c∧a)
= (a∧b∧a) ∨ a = a in A. The case where the locations of b and c are switched is similar. Finally,
one similarly verifies x∨(y∧z)∨x = (x∨y∨x) ∧ (x∨z∨x). £
We eventually show that all skew lattices in this subvariety are in fact strictly categorical
and also cancellative. Query: are skew lattices in this subvariety characterized by being
cancellative, distributive, strictly categorical and symmetric?
Example 2.4.2. [Kinyon and Leech, 2013] A minimal noncategorical skew chain has
the following Hasse diagram where a1 > b1, b3; a2 > b2, b4; b1, b2 > c1; b3, b2 > c2; and thus both
ai > both cj.
A a1 − a2
( )B b1 −A b2 −C b3 −A b4 −C b1
C c1 − c2 .
Instances of left-handed operations are given by a1 ∨ c2 = a2 = a1 ∨ a2,
a1 ∧ b4 = b3 ∧ b4 = b3 and b1 ∨ c2 = b1 ∨ b4 = b4. £
Before moving to the next section we clear up one detail:
Proposition 2.4.18. Every coset bijection ϕ: Ai → Bj is a skew lattice isomorphism.
70
Theorem 2.4.16. Normal skew lattices are strictly categorical.
Proof. Since the lower D-class in any maximal primitive subalgebra of a normal skew lattice has
exactly one coset, the composition of adjacent coset bijections is a nonempty coset bijection. £
A third class of categorical skew lattices is as follows.
Theorem 2.4.17. Every primitive skew lattice is strictly categorical, distributive and
symmetric. Thus all skew lattices in the subvariety generated from the class of primitive skew
lattices are categorical, distributive and symmetric.
Proof. Any primitive skew lattice S is trivially strictly categorical. The only nontrivial instances
of either x∨y = y∨x or x∧y = y∧x when S is primitive are when x > y or y > x. Thus S is also
symmetric. Next, consider the equation x∧(y∨z)∧x = (x∧y∧x)∨(x∧z∧x). It holds trivially when x
is in the lower D-class B. So let x = a in the upper class A and let y = b and z = c in B. Then
a∧(b∨c)∧a = a∧(c∧b)∧a = (a∧c∧a) ∧ (a∧b∧a) = (a∧b∧a) ∨ (a∧c∧a) in B.
If a in A and say b in B but c in A then b∨c in A so that a∧(b∨c)∧a = a, while (a∧b∧a) ∨ (a∧c∧a)
= (a∧b∧a) ∨ a = a in A. The case where the locations of b and c are switched is similar. Finally,
one similarly verifies x∨(y∧z)∨x = (x∨y∨x) ∧ (x∨z∨x). £
We eventually show that all skew lattices in this subvariety are in fact strictly categorical
and also cancellative. Query: are skew lattices in this subvariety characterized by being
cancellative, distributive, strictly categorical and symmetric?
Example 2.4.2. [Kinyon and Leech, 2013] A minimal noncategorical skew chain has
the following Hasse diagram where a1 > b1, b3; a2 > b2, b4; b1, b2 > c1; b3, b2 > c2; and thus both
ai > both cj.
A a1 − a2
( )B b1 −A b2 −C b3 −A b4 −C b1
C c1 − c2 .
Instances of left-handed operations are given by a1 ∨ c2 = a2 = a1 ∨ a2,
a1 ∧ b4 = b3 ∧ b4 = b3 and b1 ∨ c2 = b1 ∨ b4 = b4. £
Before moving to the next section we clear up one detail:
Proposition 2.4.18. Every coset bijection ϕ: Ai → Bj is a skew lattice isomorphism.
70
