Page 71 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 71
II: Skew Lattices
Theorem 2.4.14.
(1) A skew lattice S is categorical if and only if for all x, p, q, r with x D p ≥ q ≥ r,
(x∧r∧x) ∨ q ∨ (x∧r∧x) = [(x∧r∧x) ∨ p ∨ (x∧r∧x)] ∧ q ∧ [(x∧r∧x) ∨ p ∨ (x∧r∧x)].
(2) Categorical skew lattices form a subvariety of skew lattices.
(3) A skew lattice S is categorical if and only if its left and right factors are categorical.
Proof. Observe that in any skew lattice S, all p such that x D p arise as (u∧x∧u) ∨ x ∨ (u∧x∧u)
for u unrestricted. Observe next that for any p, all q, r such that p ≥ q ≥ r arise q = p∧v∧p and
r = q∧w∧q for v and w unrestricted. Thus the condition stated in (1) can be made unconditional
so that (2) follows from (1) and thus (3) follows from (2). To see that (1) holds, consider the case
of a nonempty composition of coset bijections ψoϕ, where ϕ is a coset bijection from A to B and
ψ is a coset bijection from B to C. Since ψoϕ is nonempty, for all elements p in A in the domain
of ψoϕ, ϕ send p to some q in B and ψ sends that q to some r in C yielding p > q > r. Clearly all
triples p > q > r arise in this fashion for some ψ and ϕ. For the given ψ and ϕ, letting χ be the
coset bijection from A to C sending p to r. Thus at least ψoϕ ⊆ χ by Lemma 4.16. The equation
of (1) states that
ψ–1[A ∧ r ∧ A] = ϕ[χ–1[A ∧ r ∧ A]]
with A∧r∧A being the image of χ. Since all indicated cosets in this equation lie in the domains
of the relevant bijections, this equation is equivalent first to [A ∧ r ∧ A] = ψoϕ[χ–1[A ∧ r ∧ A]]
and then to χ[χ–1[A∧r∧A] = ψoϕ [χ–1[A∧r∧A]] so that ψoϕ must be all of χ. £
While 2.4.16(1) at first may appear to be an obfuscation of the simple implication, if ψoϕ
⊆ χ then ψoϕ = χ, it does unpack ψ○ϕ = χ at the element-wise level, thus setting the stage for
(2) and (3). In Chapter 5, (strictly) categorical skew lattices will be studied more closely.
We next present several classes of categorical skew lattices.
Theorem 2.4.15. Skew lattices in rings are categorical.
Proof. Assume first that S is a right-handed skew lattice in a ring R and that x R p ≥ q ≥ r in S.
The conditional identity in of the previous theorem thus reduces to (r∧x) ∨ q = q ∧ [(r∧x) ∨ p].
Applying ○ and multiplication, the left side of the equation reduces to rx + q – rxq = rx + q – r,
and the right side reduces to q[rx + p – rxp] = rx + q – r again. Hence S is categorical.
Similarly, if S is left handed in R, then S must be categorical.
Next suppose that is neither left nor right-handed with either ○ or ∇ for a join. S will be
categorical if all countable subalgebras are thus. So let Sʹ be a countable subalgebra of S. By
symmetry, copies the maximal left and right-handed images of Sʹ arise as subalgebras of Sʹ in R
and thus these are categorical. By the Theorem 2.4.14(3), Sʹ is categorical. Since this holds for
all countable subalgebras of S, S is also categorical. £
69
Theorem 2.4.14.
(1) A skew lattice S is categorical if and only if for all x, p, q, r with x D p ≥ q ≥ r,
(x∧r∧x) ∨ q ∨ (x∧r∧x) = [(x∧r∧x) ∨ p ∨ (x∧r∧x)] ∧ q ∧ [(x∧r∧x) ∨ p ∨ (x∧r∧x)].
(2) Categorical skew lattices form a subvariety of skew lattices.
(3) A skew lattice S is categorical if and only if its left and right factors are categorical.
Proof. Observe that in any skew lattice S, all p such that x D p arise as (u∧x∧u) ∨ x ∨ (u∧x∧u)
for u unrestricted. Observe next that for any p, all q, r such that p ≥ q ≥ r arise q = p∧v∧p and
r = q∧w∧q for v and w unrestricted. Thus the condition stated in (1) can be made unconditional
so that (2) follows from (1) and thus (3) follows from (2). To see that (1) holds, consider the case
of a nonempty composition of coset bijections ψoϕ, where ϕ is a coset bijection from A to B and
ψ is a coset bijection from B to C. Since ψoϕ is nonempty, for all elements p in A in the domain
of ψoϕ, ϕ send p to some q in B and ψ sends that q to some r in C yielding p > q > r. Clearly all
triples p > q > r arise in this fashion for some ψ and ϕ. For the given ψ and ϕ, letting χ be the
coset bijection from A to C sending p to r. Thus at least ψoϕ ⊆ χ by Lemma 4.16. The equation
of (1) states that
ψ–1[A ∧ r ∧ A] = ϕ[χ–1[A ∧ r ∧ A]]
with A∧r∧A being the image of χ. Since all indicated cosets in this equation lie in the domains
of the relevant bijections, this equation is equivalent first to [A ∧ r ∧ A] = ψoϕ[χ–1[A ∧ r ∧ A]]
and then to χ[χ–1[A∧r∧A] = ψoϕ [χ–1[A∧r∧A]] so that ψoϕ must be all of χ. £
While 2.4.16(1) at first may appear to be an obfuscation of the simple implication, if ψoϕ
⊆ χ then ψoϕ = χ, it does unpack ψ○ϕ = χ at the element-wise level, thus setting the stage for
(2) and (3). In Chapter 5, (strictly) categorical skew lattices will be studied more closely.
We next present several classes of categorical skew lattices.
Theorem 2.4.15. Skew lattices in rings are categorical.
Proof. Assume first that S is a right-handed skew lattice in a ring R and that x R p ≥ q ≥ r in S.
The conditional identity in of the previous theorem thus reduces to (r∧x) ∨ q = q ∧ [(r∧x) ∨ p].
Applying ○ and multiplication, the left side of the equation reduces to rx + q – rxq = rx + q – r,
and the right side reduces to q[rx + p – rxp] = rx + q – r again. Hence S is categorical.
Similarly, if S is left handed in R, then S must be categorical.
Next suppose that is neither left nor right-handed with either ○ or ∇ for a join. S will be
categorical if all countable subalgebras are thus. So let Sʹ be a countable subalgebra of S. By
symmetry, copies the maximal left and right-handed images of Sʹ arise as subalgebras of Sʹ in R
and thus these are categorical. By the Theorem 2.4.14(3), Sʹ is categorical. Since this holds for
all countable subalgebras of S, S is also categorical. £
69
