Page 70 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 70
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
[A∧b∧A ∩ C∨b∨C]∨a∨[A∧b∧A ∩ C∨b∨C] ⊆ C∨b∨C∨a∨C∨b∨C = C∨a∨C in A
and
[A∧b∧A ∩ C∨b∨C]∧c∧[A∧b∧A ∩ C∨b∨C] ⊆ A∧b∧A∧c∧A∧b∧A = A∧c∧A in C,
where the equalities on the right are due to regularity. Here the expression to the left of ⊆ are the
respective input and output sets of ψoϕ. Applying the letter to any x in the input set we get
ψoϕ(x) = (x∧b∧x)∧c∧(x∧b∧x) = x∧b∧c∧b∧x = x∧c∧x = χ(x). £
A skew chain A > B > C is categorical if ψoϕ = χ always holds in (3) above, in which
case every coset bijection χ between a C-coset in A and an A-coset in C must factor as such.
Indeed, given a ∈A and c ∈C such that χ sends a to c, some b ∈B exists such that a > b > c.
(Set b = a∧(c∨y∨c)∧a for some y ∈B.) If ϕ and ψ are the A-B and B-C coset bijections sending a
to b and b to c, then χ = ψoϕ. A skew lattice S is categorical if every chain of D-classes
A > B > C in S is categorical. S is strictly categorical if it is categorical and all such composites
ψoϕ of coset bijections between cosets in comparable D-classes A > B > C are nonempty. The
outer cosets in a categorical skew chain A > B > C induce virtual versions of themselves in the
middle D-class B as follows.
Theorem 2.4.12. (Cvetko-Vah [2005c]) A skew chain A > B > C is categorical if and
only if for all a ∈ A, b ∈ B and c ∈ C that satisfy a > b > c,
(C∨a∨C) ∧ b ∧ (C∨a∨C) = (A∧b∧A) ∩ (C∨b∨C) = (A∧c∧A) ∨ b ∨ (A∧c∧A).
Proof. In terms of the coset bijections ϕ: Ai → B j , ψ: B′k → Cl and χ: A′m → C′n such that
ϕ(a) = b, ψ(b) = c and χ(a) = c with ψ○ϕ = χ, the given equalities are what is needed for ψoϕ to
occur. £
The term “categorical” comes from the following result:
Theorem 2.4.13. If a skew lattice S is strictly categorical, then a category Cat(S) is
defined by:
(1) Letting the objects of S to be the D-classes of S.
(2) For comparable D-classes A ≥ B, Hom(A, B) consists of all coset bijections from
all B-cosets in A to A-cosets in B. Otherwise, Hom(A, B) is empty.
(3) In particular, each Hom(A, A) consists of the unique identity bijection on A.
(4) Letting morphism composition be the usual composition of partial bijections.
When S is just categorical, a modified category Cat0(S) is given as above, but in addition, for
each comparable pair A ≥ B, Hom(A, B) contains a labeled copy of the empty bijection 0AB. £
68
[A∧b∧A ∩ C∨b∨C]∨a∨[A∧b∧A ∩ C∨b∨C] ⊆ C∨b∨C∨a∨C∨b∨C = C∨a∨C in A
and
[A∧b∧A ∩ C∨b∨C]∧c∧[A∧b∧A ∩ C∨b∨C] ⊆ A∧b∧A∧c∧A∧b∧A = A∧c∧A in C,
where the equalities on the right are due to regularity. Here the expression to the left of ⊆ are the
respective input and output sets of ψoϕ. Applying the letter to any x in the input set we get
ψoϕ(x) = (x∧b∧x)∧c∧(x∧b∧x) = x∧b∧c∧b∧x = x∧c∧x = χ(x). £
A skew chain A > B > C is categorical if ψoϕ = χ always holds in (3) above, in which
case every coset bijection χ between a C-coset in A and an A-coset in C must factor as such.
Indeed, given a ∈A and c ∈C such that χ sends a to c, some b ∈B exists such that a > b > c.
(Set b = a∧(c∨y∨c)∧a for some y ∈B.) If ϕ and ψ are the A-B and B-C coset bijections sending a
to b and b to c, then χ = ψoϕ. A skew lattice S is categorical if every chain of D-classes
A > B > C in S is categorical. S is strictly categorical if it is categorical and all such composites
ψoϕ of coset bijections between cosets in comparable D-classes A > B > C are nonempty. The
outer cosets in a categorical skew chain A > B > C induce virtual versions of themselves in the
middle D-class B as follows.
Theorem 2.4.12. (Cvetko-Vah [2005c]) A skew chain A > B > C is categorical if and
only if for all a ∈ A, b ∈ B and c ∈ C that satisfy a > b > c,
(C∨a∨C) ∧ b ∧ (C∨a∨C) = (A∧b∧A) ∩ (C∨b∨C) = (A∧c∧A) ∨ b ∨ (A∧c∧A).
Proof. In terms of the coset bijections ϕ: Ai → B j , ψ: B′k → Cl and χ: A′m → C′n such that
ϕ(a) = b, ψ(b) = c and χ(a) = c with ψ○ϕ = χ, the given equalities are what is needed for ψoϕ to
occur. £
The term “categorical” comes from the following result:
Theorem 2.4.13. If a skew lattice S is strictly categorical, then a category Cat(S) is
defined by:
(1) Letting the objects of S to be the D-classes of S.
(2) For comparable D-classes A ≥ B, Hom(A, B) consists of all coset bijections from
all B-cosets in A to A-cosets in B. Otherwise, Hom(A, B) is empty.
(3) In particular, each Hom(A, A) consists of the unique identity bijection on A.
(4) Letting morphism composition be the usual composition of partial bijections.
When S is just categorical, a modified category Cat0(S) is given as above, but in addition, for
each comparable pair A ≥ B, Hom(A, B) contains a labeled copy of the empty bijection 0AB. £
68
