Page 84 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 84
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Recall that the center Z(S) of S is the union of all trivial D-classes of S. Since S is normal, Z(S)
is a (possible empty) ideal of S that is isomorphic to its image in T which is also an ideal of T.
Z(S) is empty precisely when the minimal D-class of S is nontrivial. In any case, all D-classes of
S that are minimal in the complement of Z(S) correspond to join-irreducible elements in T.
Lemma 2.6.10. Given S and T as above, let X be a minimal D-class in the complement
of Z(S), let x be fixed in X and let P be the prime filter in T induced by the image of X in π. Set
Sʹ = (S \ S∨x∨S) ∪ x∨S∨x
and let T[X⎮P] be the P-primary algebra induced by X and P. Then
i) Sʹ is a subalgbra of S that is also mapped onto T by the canonical epimorphism from S.
ii) An isomorphism θ: S ≅ Sʹ ×T T[X⎮P] is given by the rule
θ(y) = ⎧(x ∨ y ∨ x, y ∧ x ∧ y) for all y ∈ S ∨ x ∨ S .
⎨⎩ y otherwise
iii) Upon comparison in T, Z(Sʹ) is properly larger than Z(S).
Proof. Sʹ is a subalgebra since x commutes with all elements in the complement of S∨x∨S. Thus
θ is at least an isomorphism off of S∨x∨S and by Theorem 2.6.3, θ also yields an isomorphism of
S∨x∨S with (x∨S∨x) × X. Suppose that u ∈ S∨x∨S and w ∈ S \ S∨x∨S are given. Then
θ(u∨w) = θ(u)∨θ(w) is equivalent to x∨u∨w∨x = x∨u∨x∨w and (u∨w)∧x∧(u∨w) = (u∧x∧u).
Since x commutes with w, the first identity holds. Because x∧w lies in Z(S), x∧u ≥ x∧w and
x∧(u∨w) = x∧u. Similarly, (u∨w)∧x = u∧x and the second identity holds. Finally,
θ(u∧w) = θ(u)∧θ(w) is equivalent to u∧w = (x∨u∨x)∧w. Distribution in the symmetric, normal
case yields
(x∨u∨x)∧w = (x∧w) ∨ (u∧w) ∨ (x∧w) = u∧w
since u∧w ≻ x∧w with x∧w in Z(S) implies u∧w ≥ x∧w. £
As a consequence we have the following Primary Decomposition Theorem.
Theorem 2.6.11. Let S be a strongly distributive skew lattice with finite maximal lattice
image T and let Pr(T) be the set of prime filters of T including T. Then to each P in Pr(T) there
corresponds to a rectangular algebra XP, that is unique to within isomorphism, such that S is
isomorphic to the fibered product ∏T{T[XP⎮P]⎮P ∈Pr(T)}. S is reduced if and only if XP is
nontrivial for each proper prime filter P.
82
Recall that the center Z(S) of S is the union of all trivial D-classes of S. Since S is normal, Z(S)
is a (possible empty) ideal of S that is isomorphic to its image in T which is also an ideal of T.
Z(S) is empty precisely when the minimal D-class of S is nontrivial. In any case, all D-classes of
S that are minimal in the complement of Z(S) correspond to join-irreducible elements in T.
Lemma 2.6.10. Given S and T as above, let X be a minimal D-class in the complement
of Z(S), let x be fixed in X and let P be the prime filter in T induced by the image of X in π. Set
Sʹ = (S \ S∨x∨S) ∪ x∨S∨x
and let T[X⎮P] be the P-primary algebra induced by X and P. Then
i) Sʹ is a subalgbra of S that is also mapped onto T by the canonical epimorphism from S.
ii) An isomorphism θ: S ≅ Sʹ ×T T[X⎮P] is given by the rule
θ(y) = ⎧(x ∨ y ∨ x, y ∧ x ∧ y) for all y ∈ S ∨ x ∨ S .
⎨⎩ y otherwise
iii) Upon comparison in T, Z(Sʹ) is properly larger than Z(S).
Proof. Sʹ is a subalgebra since x commutes with all elements in the complement of S∨x∨S. Thus
θ is at least an isomorphism off of S∨x∨S and by Theorem 2.6.3, θ also yields an isomorphism of
S∨x∨S with (x∨S∨x) × X. Suppose that u ∈ S∨x∨S and w ∈ S \ S∨x∨S are given. Then
θ(u∨w) = θ(u)∨θ(w) is equivalent to x∨u∨w∨x = x∨u∨x∨w and (u∨w)∧x∧(u∨w) = (u∧x∧u).
Since x commutes with w, the first identity holds. Because x∧w lies in Z(S), x∧u ≥ x∧w and
x∧(u∨w) = x∧u. Similarly, (u∨w)∧x = u∧x and the second identity holds. Finally,
θ(u∧w) = θ(u)∧θ(w) is equivalent to u∧w = (x∨u∨x)∧w. Distribution in the symmetric, normal
case yields
(x∨u∨x)∧w = (x∧w) ∨ (u∧w) ∨ (x∧w) = u∧w
since u∧w ≻ x∧w with x∧w in Z(S) implies u∧w ≥ x∧w. £
As a consequence we have the following Primary Decomposition Theorem.
Theorem 2.6.11. Let S be a strongly distributive skew lattice with finite maximal lattice
image T and let Pr(T) be the set of prime filters of T including T. Then to each P in Pr(T) there
corresponds to a rectangular algebra XP, that is unique to within isomorphism, such that S is
isomorphic to the fibered product ∏T{T[XP⎮P]⎮P ∈Pr(T)}. S is reduced if and only if XP is
nontrivial for each proper prime filter P.
82
