Page 144 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 144
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 4.4.8. Given a skew Boolean ∩-algebra S, its congruence lattice Con∩(S) on is
isomorphic to the lattice of ideals of S (or of its maximal generalized Boolean image S/D.).
Finitely generated congruences correspond to the principal ideals of S/D and thus form a
generalized Boolean sublattice of the congruence lattice.
Proof. The general correspondence is clear by the previous lemma. By the lemma again, finitely
generated congruences have finitely generated ideal kernels that must be principal. Conversely, a
principal ideal 〈a〉 corresponds to the congruence generated from (a, 0). £
Theorem 4.4.9. The lattice Con(S) of all skew lattice congruences on a skew Boolean
∩-algebra S is the subdirect product of the interval [Δ, D] and the sublattice Con∩(S) of skew
Boolean ∩-algebra congruences on S under the map Con(S) → [Δ, D] × Con∩(S) given by the
rule θ → (θ∩D, θθ[0]).
Proof. By Theorem 3.1.2, Con(S) is the subdirect product of [Δ, D] and the interval [D, ∇]
under the map θ → (θ∩D, θ∨D). But [D, ∇] ≅ Con(S/D) which is isomorphic to the lattice of
ideals of S/D and in turn to the lattice of ideals of S, and thus to Con∩(S). The ∩-respecting
congruence corresponding to θ∨D has kernel ideal (θ∨D)(0) = θ(0) so that θ(θ∨D)(0) = θθ(0). £
Intersections and the canonical images of a skew lattice
Given a skew lattice S, the canonical skew lattice maps, S → S/L, S → S/R and S → S/D,
are all homomorphisms. How do intersections fit into this picture? We begin with:
Theorem 4.4.10. Given a skew lattice S with finite intersections:
i) The canonical map S → S/L preserves intersections iff S is right-handed, so that S/L ≅ S.
ii) The canonical map S → S/R preserves intersections iff S is left-handed, so that S/R ≅ S.
iii) Both maps preserve finite intersections if and only if S is a lattice, in which case ∩ is ∧.
iv) The canonical map S → S/D preserves intersections iff S is a lattice, so that S/D ≅ S.
Proof. Suppose the map S → S/L preserves finite intersections and let x and y be distinct in S,
but L-related. Then x∩y exists in a properly lower D-class in S, while their images merge in S/L,
giving them a trivial intersection in the D-class of S/L corresponding to that of both x and y in S.
Thus (i) is seen. That cases for (ii) and (iv) are similar and (iii) follows immediately. £
Since neither S/L nor S/R need be isomorphic to S in general, they need not, in general,
inherit intersections. This raises questions: Under what added conditions must a skew lattice S
142
Theorem 4.4.8. Given a skew Boolean ∩-algebra S, its congruence lattice Con∩(S) on is
isomorphic to the lattice of ideals of S (or of its maximal generalized Boolean image S/D.).
Finitely generated congruences correspond to the principal ideals of S/D and thus form a
generalized Boolean sublattice of the congruence lattice.
Proof. The general correspondence is clear by the previous lemma. By the lemma again, finitely
generated congruences have finitely generated ideal kernels that must be principal. Conversely, a
principal ideal 〈a〉 corresponds to the congruence generated from (a, 0). £
Theorem 4.4.9. The lattice Con(S) of all skew lattice congruences on a skew Boolean
∩-algebra S is the subdirect product of the interval [Δ, D] and the sublattice Con∩(S) of skew
Boolean ∩-algebra congruences on S under the map Con(S) → [Δ, D] × Con∩(S) given by the
rule θ → (θ∩D, θθ[0]).
Proof. By Theorem 3.1.2, Con(S) is the subdirect product of [Δ, D] and the interval [D, ∇]
under the map θ → (θ∩D, θ∨D). But [D, ∇] ≅ Con(S/D) which is isomorphic to the lattice of
ideals of S/D and in turn to the lattice of ideals of S, and thus to Con∩(S). The ∩-respecting
congruence corresponding to θ∨D has kernel ideal (θ∨D)(0) = θ(0) so that θ(θ∨D)(0) = θθ(0). £
Intersections and the canonical images of a skew lattice
Given a skew lattice S, the canonical skew lattice maps, S → S/L, S → S/R and S → S/D,
are all homomorphisms. How do intersections fit into this picture? We begin with:
Theorem 4.4.10. Given a skew lattice S with finite intersections:
i) The canonical map S → S/L preserves intersections iff S is right-handed, so that S/L ≅ S.
ii) The canonical map S → S/R preserves intersections iff S is left-handed, so that S/R ≅ S.
iii) Both maps preserve finite intersections if and only if S is a lattice, in which case ∩ is ∧.
iv) The canonical map S → S/D preserves intersections iff S is a lattice, so that S/D ≅ S.
Proof. Suppose the map S → S/L preserves finite intersections and let x and y be distinct in S,
but L-related. Then x∩y exists in a properly lower D-class in S, while their images merge in S/L,
giving them a trivial intersection in the D-class of S/L corresponding to that of both x and y in S.
Thus (i) is seen. That cases for (ii) and (iv) are similar and (iii) follows immediately. £
Since neither S/L nor S/R need be isomorphic to S in general, they need not, in general,
inherit intersections. This raises questions: Under what added conditions must a skew lattice S
142
