Page 100 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Given a regular quasilattice Q, let Q(l, l), Q(l, r), Q(r, l), and Q(r, r) denote the maximal flat
images of each type. Define Φ: Q → Q(l, l) × Q(l, r) × Q(r, l) × Q(r, r) by:
Φ(x) = (xϕ(l, l), xϕ(l, r), xϕ(r, l), xϕ(r, r)).
Φ is a monomorphism, since the involved congruences intersect to Δ. To describe its image,
observe that Q shares with its four maximal flat images a common maximal lattice image, Q/D.
If δ(u,v): Q(u,v) → Q/D, for (u, v) ∈ {(l, l), (l, r), (r, l), (r, r)}, are epimorphisms induced by the
inclusions of the ϕ(u, v) into D, then Φ(Q) lies in
{(u, v, w, y) ∈ Q(l, l) × Q(l, r) × Q(r, l) × Q(r, r)⎮δ(l, l)(u) = δ(l, r)(v) = δ(r, l)(w) = δ(r, r)(y)},
the fibered product over Q/D of the various maximal flat images of Q. In fact:
Theorem 3.3.4. If Q is a regular quasilattice, then Φ is an isomorphism of Q with the
fibered product over Q/D of the four maximal flat images of Q.
Q ≅ Q(l, l) ×Q/D Q(l, r) ×Q/D Q(r, l) ×Q/D Q(r, r)
Proof. Restricting our attention to each D-class A of Q, the respective rectangular band
structures yield first A ≅ A/R(∨) × A/L(∨) and then in turn
A/R(∨) ≅ A/R(∨)/R(∧) × A/R(∨)/L(∧) ≅ A/R(∨)oR(∧) × A/R(∨)oL(∧) = A/ϕ(l, l) × A/ϕ(l, r)
with A/L(∨) similarly factoring as A/ϕ(r, l) × A/ϕ(r, r). Thus class A factors as a product of its
maximal flat images. In passing from D-classes to Q, the result follows. £
The above theorem is the quasilattice version of a result of Kimura [1958] about regular
bands. For a general discussion of permuting congruences and fibered products, see Grätzer
[1979].
3.4 Paralattices and refined quasilattices
Skew lattices are simultaneously quasilattices and paralattices and thus possess the
features of both. While quasilattices possess a coherent Clifford-McLean structure, paralattices
need not. Nor need (flat) paralattices be regular. Thus some of our basic intuitions about bands
and quasilattices no longer always hold. We begin our look at paralattices with a special case.
Unlike skew lattices and more generally quasilattices, it is possible for say ∨ to be
commutative, but not ∧. (For quasilattices, if say ∨ is commutative, then D∨ = Δ and hence
D∧ = Δ, making ∧ commutative.) A band with joins, or a ∨-band, is a band B whose natural
98
Given a regular quasilattice Q, let Q(l, l), Q(l, r), Q(r, l), and Q(r, r) denote the maximal flat
images of each type. Define Φ: Q → Q(l, l) × Q(l, r) × Q(r, l) × Q(r, r) by:
Φ(x) = (xϕ(l, l), xϕ(l, r), xϕ(r, l), xϕ(r, r)).
Φ is a monomorphism, since the involved congruences intersect to Δ. To describe its image,
observe that Q shares with its four maximal flat images a common maximal lattice image, Q/D.
If δ(u,v): Q(u,v) → Q/D, for (u, v) ∈ {(l, l), (l, r), (r, l), (r, r)}, are epimorphisms induced by the
inclusions of the ϕ(u, v) into D, then Φ(Q) lies in
{(u, v, w, y) ∈ Q(l, l) × Q(l, r) × Q(r, l) × Q(r, r)⎮δ(l, l)(u) = δ(l, r)(v) = δ(r, l)(w) = δ(r, r)(y)},
the fibered product over Q/D of the various maximal flat images of Q. In fact:
Theorem 3.3.4. If Q is a regular quasilattice, then Φ is an isomorphism of Q with the
fibered product over Q/D of the four maximal flat images of Q.
Q ≅ Q(l, l) ×Q/D Q(l, r) ×Q/D Q(r, l) ×Q/D Q(r, r)
Proof. Restricting our attention to each D-class A of Q, the respective rectangular band
structures yield first A ≅ A/R(∨) × A/L(∨) and then in turn
A/R(∨) ≅ A/R(∨)/R(∧) × A/R(∨)/L(∧) ≅ A/R(∨)oR(∧) × A/R(∨)oL(∧) = A/ϕ(l, l) × A/ϕ(l, r)
with A/L(∨) similarly factoring as A/ϕ(r, l) × A/ϕ(r, r). Thus class A factors as a product of its
maximal flat images. In passing from D-classes to Q, the result follows. £
The above theorem is the quasilattice version of a result of Kimura [1958] about regular
bands. For a general discussion of permuting congruences and fibered products, see Grätzer
[1979].
3.4 Paralattices and refined quasilattices
Skew lattices are simultaneously quasilattices and paralattices and thus possess the
features of both. While quasilattices possess a coherent Clifford-McLean structure, paralattices
need not. Nor need (flat) paralattices be regular. Thus some of our basic intuitions about bands
and quasilattices no longer always hold. We begin our look at paralattices with a special case.
Unlike skew lattices and more generally quasilattices, it is possible for say ∨ to be
commutative, but not ∧. (For quasilattices, if say ∨ is commutative, then D∨ = Δ and hence
D∧ = Δ, making ∧ commutative.) A band with joins, or a ∨-band, is a band B whose natural
98
