Page 180 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 180
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
The answer to the first question is, no. Primitive skew lattices are trivially categorical,
and thus all skew lattices in the first variety are categorical. We shall see in the next section that
skew chains A > B > C of length 2 exist that are not categorical. The first variety is thus a proper
subvariety of the second.
5.4 Categorical skew lattices
We continue our study begun in Section 2.4 of categorical skew lattices, where
all nonempty composites of coset bijections are coset bijections. This reduces to the implication:
if a > b > c with aʹ D a and cʹ D c such that cʹ = aʹ∧c∧aʹ in C and aʹ = cʹ∨a∨cʹ in A (making both
a > c and aʹ > cʹ part of a common coset bijection from A to C), then aʹ∧b∧aʹ = cʹ∨b∨cʹ in B.
a – a′ = c′ ∨ a ∨ c′
b – b′ = a′ ∧ b ∧ a′ = c′ ∨ b ∨ c′ (where denotes >)
c – c′ = a′ ∧ c ∧ a′
For if χ, the unique coset bijection from A to C taking a to c, factors as ψοϕ, where ϕ and ψ are
the unique coset bijections from A to B and from B to C taking a to b and b to c respectively, then
one has aʹ∧b∧aʹ = ϕ[aʹ] = ψ–1[cʹ] = cʹ∨b∨cʹ.
Contraposition gives the following criterion for a skew lattice S to not be categorical:
given a > b > c in S with aʹ D a and cʹ D c being such that aʹ∧c∧aʹ = cʹ in Dc and cʹ∨a∨cʹ = aʹ in
Da, but aʹ∧b∧aʹ ≠ cʹ∨b∨cʹin Db.
We have seen that categorical skew lattices form a variety. In the left-handed skew case:
Lemma 5.4.1. A left-handed skew lattice is categorical if either of the following pair of
dual identities hold:
x ∧ [y ∨ (x ∧ y ∧ z)] = x ∧ y (5.4.1)
or
[(x ∨ y ∨ z) ∧ y] ∨ z = y ∨ z. (5.4.2)
Proof. Let S be a left-handed noncategorical skew lattice. Thus a > b > c in S exist with aʹ L a
and cʹ L c such that aʹ ∧ c = cʹ in C, a ∨ cʹ = aʹ in A but aʹ ∧ b ≠ b ∨ cʹ in B. This creates the
following configuration
A: a − a′
B: b − a′ ∧ b − b ∨ c′ .
C: c − c′
178
The answer to the first question is, no. Primitive skew lattices are trivially categorical,
and thus all skew lattices in the first variety are categorical. We shall see in the next section that
skew chains A > B > C of length 2 exist that are not categorical. The first variety is thus a proper
subvariety of the second.
5.4 Categorical skew lattices
We continue our study begun in Section 2.4 of categorical skew lattices, where
all nonempty composites of coset bijections are coset bijections. This reduces to the implication:
if a > b > c with aʹ D a and cʹ D c such that cʹ = aʹ∧c∧aʹ in C and aʹ = cʹ∨a∨cʹ in A (making both
a > c and aʹ > cʹ part of a common coset bijection from A to C), then aʹ∧b∧aʹ = cʹ∨b∨cʹ in B.
a – a′ = c′ ∨ a ∨ c′
b – b′ = a′ ∧ b ∧ a′ = c′ ∨ b ∨ c′ (where denotes >)
c – c′ = a′ ∧ c ∧ a′
For if χ, the unique coset bijection from A to C taking a to c, factors as ψοϕ, where ϕ and ψ are
the unique coset bijections from A to B and from B to C taking a to b and b to c respectively, then
one has aʹ∧b∧aʹ = ϕ[aʹ] = ψ–1[cʹ] = cʹ∨b∨cʹ.
Contraposition gives the following criterion for a skew lattice S to not be categorical:
given a > b > c in S with aʹ D a and cʹ D c being such that aʹ∧c∧aʹ = cʹ in Dc and cʹ∨a∨cʹ = aʹ in
Da, but aʹ∧b∧aʹ ≠ cʹ∨b∨cʹin Db.
We have seen that categorical skew lattices form a variety. In the left-handed skew case:
Lemma 5.4.1. A left-handed skew lattice is categorical if either of the following pair of
dual identities hold:
x ∧ [y ∨ (x ∧ y ∧ z)] = x ∧ y (5.4.1)
or
[(x ∨ y ∨ z) ∧ y] ∨ z = y ∨ z. (5.4.2)
Proof. Let S be a left-handed noncategorical skew lattice. Thus a > b > c in S exist with aʹ L a
and cʹ L c such that aʹ ∧ c = cʹ in C, a ∨ cʹ = aʹ in A but aʹ ∧ b ≠ b ∨ cʹ in B. This creates the
following configuration
A: a − a′
B: b − a′ ∧ b − b ∨ c′ .
C: c − c′
178
