Page 143 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 143
IV: Skew Boolean Algebras
Theorem 4.4.6. In the variety of skew Boolean ∩-algebras the following assertions hold.
i) The primitive algebras are precisely the nontrivial simple algebras.
ii) The primitive algebras are the nontrivial directly irreducible algebras.
iii) The primitive algebras are the nontrivial subdirectly irreducible algebras.
Proof. By Theorem 4.4.5, a nontrivial simple algebra must be primitive. Conversely, let S be a
primitive algebra and let θ be a congruence on S. Suppose that e θ f with e ≠ f in S. Then e and f
are also θ-congruent to e∩f = 0. Since either e ≠ 0 or f ≠ 0, this forces every element of the
primitive algebra to be congruent to 0. Hence θ is the universal congruence. The only other
congruence possible is thus the identity congruence Δ. Thus all primitive algebras are simple and
(i) holds. (ii) is clear. Given (i) and (ii), (iii) easily follows. £
Given a primitive skew Boolean algebra P and an ∩-preserving homomorphism ϕ from P
to a skew Boolean ∩-algebra S, it follows that ϕ is either the 0-homomorpism, or an embedding.
The ideals of any skew lattice S form a complete lattice. The canonical map π: S → S/D
induces an isomorphism of this lattice with the lattice of ideals of S/D, via the map
I → I/D = S/D. If S/D is finite, then all ideals of both S and S/D are principal with both lattices
of ideals being isomorphic to S/D. In the current context, ideals serve as the kernels of
homomorphism.
Lemma 4.4.7. Let θ be a congruence on a skew Boolean ∩-algebra S and let θ[0] be the
congruence class of 0. Then the class θ[0] is an ideal of S and for all e, f ∈ S,
e θ f if and only if (e \ e∩f) ∨ (f \ e∩f) ∈ θ[0].
Conversely, if I is an ideal of S, then the relation θ = θI defined as above is a congruence on S.
Moreover the maps θ → θ[0] and I → θI form an inverse pair of bijections.
Proof. Observe that on any skew Boolean ∩-algebra, e = f if and only if (e \ e∩f) ∨ (f \ e∩f) = 0.
This is certainly true in the primitive case. Thus it is true for all such algebras by Theorem 4.3.6
and the first assertion of the lemma follows. For the converse, first let I be a principal ideal 〈a〉
with complement ann(a). By Theorem 4.4.5, an ∩-algebra homomorphism ϕ of S onto ann(a)
exists for which 〈a〉 = ϕ–1(0). If θ is the congruence associated with ϕ, then
e θ f in S iff ϕ(e) = ϕ(f) in ann(a)
iff (ϕ(e) \ ϕ(e)∩ϕ(f)) ∨ (ϕ(f) \ ϕ(e)∩ϕ(f)) = 0 in ann(a)
iff (e \ e∩f) ∨ (f \ e∩f) ∈ 〈a〉.
In general, any ideal I is the directed union of principal ideals: I = ∪↑{〈a〉⎪a ∈ I}. The θI as
defined above is the corresponding directed union ∪↑{θ〈a〉⎪a ∈ I} of “principal congruences” and
thus is also a congruence. That both processes are reciprocal is clear. £
141
Theorem 4.4.6. In the variety of skew Boolean ∩-algebras the following assertions hold.
i) The primitive algebras are precisely the nontrivial simple algebras.
ii) The primitive algebras are the nontrivial directly irreducible algebras.
iii) The primitive algebras are the nontrivial subdirectly irreducible algebras.
Proof. By Theorem 4.4.5, a nontrivial simple algebra must be primitive. Conversely, let S be a
primitive algebra and let θ be a congruence on S. Suppose that e θ f with e ≠ f in S. Then e and f
are also θ-congruent to e∩f = 0. Since either e ≠ 0 or f ≠ 0, this forces every element of the
primitive algebra to be congruent to 0. Hence θ is the universal congruence. The only other
congruence possible is thus the identity congruence Δ. Thus all primitive algebras are simple and
(i) holds. (ii) is clear. Given (i) and (ii), (iii) easily follows. £
Given a primitive skew Boolean algebra P and an ∩-preserving homomorphism ϕ from P
to a skew Boolean ∩-algebra S, it follows that ϕ is either the 0-homomorpism, or an embedding.
The ideals of any skew lattice S form a complete lattice. The canonical map π: S → S/D
induces an isomorphism of this lattice with the lattice of ideals of S/D, via the map
I → I/D = S/D. If S/D is finite, then all ideals of both S and S/D are principal with both lattices
of ideals being isomorphic to S/D. In the current context, ideals serve as the kernels of
homomorphism.
Lemma 4.4.7. Let θ be a congruence on a skew Boolean ∩-algebra S and let θ[0] be the
congruence class of 0. Then the class θ[0] is an ideal of S and for all e, f ∈ S,
e θ f if and only if (e \ e∩f) ∨ (f \ e∩f) ∈ θ[0].
Conversely, if I is an ideal of S, then the relation θ = θI defined as above is a congruence on S.
Moreover the maps θ → θ[0] and I → θI form an inverse pair of bijections.
Proof. Observe that on any skew Boolean ∩-algebra, e = f if and only if (e \ e∩f) ∨ (f \ e∩f) = 0.
This is certainly true in the primitive case. Thus it is true for all such algebras by Theorem 4.3.6
and the first assertion of the lemma follows. For the converse, first let I be a principal ideal 〈a〉
with complement ann(a). By Theorem 4.4.5, an ∩-algebra homomorphism ϕ of S onto ann(a)
exists for which 〈a〉 = ϕ–1(0). If θ is the congruence associated with ϕ, then
e θ f in S iff ϕ(e) = ϕ(f) in ann(a)
iff (ϕ(e) \ ϕ(e)∩ϕ(f)) ∨ (ϕ(f) \ ϕ(e)∩ϕ(f)) = 0 in ann(a)
iff (e \ e∩f) ∨ (f \ e∩f) ∈ 〈a〉.
In general, any ideal I is the directed union of principal ideals: I = ∪↑{〈a〉⎪a ∈ I}. The θI as
defined above is the corresponding directed union ∪↑{θ〈a〉⎪a ∈ I} of “principal congruences” and
thus is also a congruence. That both processes are reciprocal is clear. £
141
