Page 81 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 81
II: Skew Lattices
join class factors as A × B with both projections being canonical: a∨b →a and a∨b →b. The
isomorphism of A0 × B0 with SA∪B is given by θ(x, y) = x∨y for x in A0 and y in B0. £
All results in this section thus far involve decompositions. We attempt to develop this
theme in what follows. We begin by generalizing the construction of Theorem 2.6.3.
Let T be a lattice and let P be a prime filter of T. Thus P is a filter and T \ P is an ideal of
T. Put otherwise:
(1) For all p ∈ P and t ∈ T, p ∨ t ∈ P.
(2) P is closed under ∧.
(3) If s ∨ t ∈ P, then either s ∈ P or t ∈ P.
Given T, P and a rectangular skew lattice X, let T[X⎮P] be the normal skew lattice defined on
(P × X) ∪ (T \ P)
by extending the operations on the skew lattice s P × X and the lattice ideal T \ P by setting
s ∨ (p, x) = (s∨p, x) = (p, x) ∨ s and s ∧ (p, x) = s∧p = (p, x) ∧ s
for (p, x) ∈ P × X and s ∈ S \ P. Any skew lattice isomorphic to T[X⎮P] is said to be P-primary
over T with fiber X. Prime filters arise as inverse images f–1(1) for lattice epimorphisms f: T → 2
where 2 is the lattice 10. Thus T[X⎮P] may be viewed as the fibered product T ×2 X0 obtained by
pulling the surjection X0 → 2 back along f: T → 2.
More generally, let P(T) be the family of all prime filters of T, including T and let
{XP⎮P ∈Pr(T)} be a corresponding family of rectangular algebras. Then the fibered product
over T, ∏T{T[XP | P]⎮P ∈P(T)} is both symmetric and normal. To within isomorphism, its
rectangular D-classes are given by setting D(t) = ∏{XP⎮P ∈Pr(T) & t ∈P} and using canonical
coordinate projections and injections. Any skew lattice S isomorphic to such a fibered product is
decomposable with the fibered product being its primary decomposition. The rest of this section
is devoted to proving a main result of this section.
Theorem 2.6.5. (The Decomposition Theorem) Every symmetric normal skew lattice
with a finitely generated maximal lattice image is decomposable. More generally, a symmetric
normal skew lattice with a finite maximal distributive lattice image is decomposable. £
The first major step in proving this result is our next theorem. But we first need several
preliminary lemmas, beginning with:
Lemma 2.6.6. Let S be a symmetric, normal skew lattice and let A ≥ B be D-classes in
S. If ≥ induces an isomorphism between A and B as rectangular skew lattices (making both full
cosets of each other), then for any D-class C such that A ≥ C ≥ B, ≥ also induces isomorphisms
between A and C and between C and B.
79
join class factors as A × B with both projections being canonical: a∨b →a and a∨b →b. The
isomorphism of A0 × B0 with SA∪B is given by θ(x, y) = x∨y for x in A0 and y in B0. £
All results in this section thus far involve decompositions. We attempt to develop this
theme in what follows. We begin by generalizing the construction of Theorem 2.6.3.
Let T be a lattice and let P be a prime filter of T. Thus P is a filter and T \ P is an ideal of
T. Put otherwise:
(1) For all p ∈ P and t ∈ T, p ∨ t ∈ P.
(2) P is closed under ∧.
(3) If s ∨ t ∈ P, then either s ∈ P or t ∈ P.
Given T, P and a rectangular skew lattice X, let T[X⎮P] be the normal skew lattice defined on
(P × X) ∪ (T \ P)
by extending the operations on the skew lattice s P × X and the lattice ideal T \ P by setting
s ∨ (p, x) = (s∨p, x) = (p, x) ∨ s and s ∧ (p, x) = s∧p = (p, x) ∧ s
for (p, x) ∈ P × X and s ∈ S \ P. Any skew lattice isomorphic to T[X⎮P] is said to be P-primary
over T with fiber X. Prime filters arise as inverse images f–1(1) for lattice epimorphisms f: T → 2
where 2 is the lattice 10. Thus T[X⎮P] may be viewed as the fibered product T ×2 X0 obtained by
pulling the surjection X0 → 2 back along f: T → 2.
More generally, let P(T) be the family of all prime filters of T, including T and let
{XP⎮P ∈Pr(T)} be a corresponding family of rectangular algebras. Then the fibered product
over T, ∏T{T[XP | P]⎮P ∈P(T)} is both symmetric and normal. To within isomorphism, its
rectangular D-classes are given by setting D(t) = ∏{XP⎮P ∈Pr(T) & t ∈P} and using canonical
coordinate projections and injections. Any skew lattice S isomorphic to such a fibered product is
decomposable with the fibered product being its primary decomposition. The rest of this section
is devoted to proving a main result of this section.
Theorem 2.6.5. (The Decomposition Theorem) Every symmetric normal skew lattice
with a finitely generated maximal lattice image is decomposable. More generally, a symmetric
normal skew lattice with a finite maximal distributive lattice image is decomposable. £
The first major step in proving this result is our next theorem. But we first need several
preliminary lemmas, beginning with:
Lemma 2.6.6. Let S be a symmetric, normal skew lattice and let A ≥ B be D-classes in
S. If ≥ induces an isomorphism between A and B as rectangular skew lattices (making both full
cosets of each other), then for any D-class C such that A ≥ C ≥ B, ≥ also induces isomorphisms
between A and C and between C and B.
79
