Page 125 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 125
IV: Skew Boolean Algebras
Proof. As stated above, (i) holds for all distributive skew lattices with zero elements. From v∧b
= 0 = b∧v for all v ∈ann(A) and all b ∈Da, (iii) follows by Theorem 2.12. Next, (ii) follows from
(iii) and symmetry. Given (e1, e2), (f1, f2) ∈(A) × ann(A), (e1∨e2) ∨ (f1∨f2) = (e1∨f1) ∨ (e2∨f2)
follows from (ii). By Theorem 2.3.4, (e1 ∨ e2) ∧ (f1 ∨ f2) expands to
(e1 ∧ f1) ∨ (e1 ∧ f2) ∨ (e2 ∧ f1) ∨ (e2 ∧ f2) = (e1 ∧ f1) ∨ (e2 ∧ f2).
Hence µ is at least a homomorphism. For any s ∈S, the decomposition s = (s∧a∧s) ∨ (s \ a)
represents s as the join of an element in S1 with an element in S2. Hence µ is also “onto”.
Finally, suppose that (e1 ∨ e2) = (f1 ∨ f2) for e1, f1 in S1 and e2, f2 in S2. Letting u be this common
join we have u ≥ e1, f1, e2 and f2. Thus
e1 = e1∧u = e1 ∧ (f1 ∨ f2) = (e1∧f1) ∨ (e1∧f2) = (e1∧f1) ∨ 0 = e1∧f1
and in similar fashion f1 = e1∧f1. Likewise e2 = f2 so that µ is indeed an isomorphism. £
Corollary 4.1.5. Every skew Boolean algebra S with a finite maximal lattice image is
isomorphic to a product of primitive skew Boolean algebras that are determined to within
isomorphism by its minimal non-0 D-classes. £
Recall that 2 is the Boolean lattice {1 > 0}, 3L is the left-handed primitive skew Boolean
algebra {1 L 2 > 0}, 3R is its right-handed variant and 5 is the fibered product, 3L ×2 3R.
Corollary 4.1.6. The nontrivial directly irreducible skew Boolean algebras are the
primitive algebras. The nontrivial subdirectly irreducible skew Boolean algebras are 2, 3L and
3R. Thus every skew Boolean algebra is a subdirect product of copies of 2, 3L and 3R. £
Proof. The first statement is clear by Theorem 4.1.4. A nontrivial, subdirectly irreducible
algebra is thus primitive, and must be either left-or right handed, thanks to the factorization of
Theorem 2.1.5. If both, it is a copy of 2. Otherwise, it is a copy of 3L or 3R by Theorem 2.6.12.
£
Corollary 4.1.7. Every skew Boolean algebra can be embedded in a power of 5. Every
left-handed (right-handed) skew Boolean algebra can be embedded in a power of 3L (of 3R).
Alternatively, every right-handed skew Boolean algebra can be embedded in some algebra of
partial functions with codomain {1, 2}. £
Corollary 4.1.8. A (quasi-)identity in ∨, ∧ and \ holds for all skew Boolean algebras if
and only if it holds on 5. It holds for all left-handed (right-handed) skew Boolean algebras if and
only if it holds on 3L (or 3R). The question of when a (quasi-)identity holds in any of these
varieties is thus decidable. £
The above results reveal the following simple lattice. Here 〈A〉 denotes the subvariety
generated by the algebra A.
123
Proof. As stated above, (i) holds for all distributive skew lattices with zero elements. From v∧b
= 0 = b∧v for all v ∈ann(A) and all b ∈Da, (iii) follows by Theorem 2.12. Next, (ii) follows from
(iii) and symmetry. Given (e1, e2), (f1, f2) ∈(A) × ann(A), (e1∨e2) ∨ (f1∨f2) = (e1∨f1) ∨ (e2∨f2)
follows from (ii). By Theorem 2.3.4, (e1 ∨ e2) ∧ (f1 ∨ f2) expands to
(e1 ∧ f1) ∨ (e1 ∧ f2) ∨ (e2 ∧ f1) ∨ (e2 ∧ f2) = (e1 ∧ f1) ∨ (e2 ∧ f2).
Hence µ is at least a homomorphism. For any s ∈S, the decomposition s = (s∧a∧s) ∨ (s \ a)
represents s as the join of an element in S1 with an element in S2. Hence µ is also “onto”.
Finally, suppose that (e1 ∨ e2) = (f1 ∨ f2) for e1, f1 in S1 and e2, f2 in S2. Letting u be this common
join we have u ≥ e1, f1, e2 and f2. Thus
e1 = e1∧u = e1 ∧ (f1 ∨ f2) = (e1∧f1) ∨ (e1∧f2) = (e1∧f1) ∨ 0 = e1∧f1
and in similar fashion f1 = e1∧f1. Likewise e2 = f2 so that µ is indeed an isomorphism. £
Corollary 4.1.5. Every skew Boolean algebra S with a finite maximal lattice image is
isomorphic to a product of primitive skew Boolean algebras that are determined to within
isomorphism by its minimal non-0 D-classes. £
Recall that 2 is the Boolean lattice {1 > 0}, 3L is the left-handed primitive skew Boolean
algebra {1 L 2 > 0}, 3R is its right-handed variant and 5 is the fibered product, 3L ×2 3R.
Corollary 4.1.6. The nontrivial directly irreducible skew Boolean algebras are the
primitive algebras. The nontrivial subdirectly irreducible skew Boolean algebras are 2, 3L and
3R. Thus every skew Boolean algebra is a subdirect product of copies of 2, 3L and 3R. £
Proof. The first statement is clear by Theorem 4.1.4. A nontrivial, subdirectly irreducible
algebra is thus primitive, and must be either left-or right handed, thanks to the factorization of
Theorem 2.1.5. If both, it is a copy of 2. Otherwise, it is a copy of 3L or 3R by Theorem 2.6.12.
£
Corollary 4.1.7. Every skew Boolean algebra can be embedded in a power of 5. Every
left-handed (right-handed) skew Boolean algebra can be embedded in a power of 3L (of 3R).
Alternatively, every right-handed skew Boolean algebra can be embedded in some algebra of
partial functions with codomain {1, 2}. £
Corollary 4.1.8. A (quasi-)identity in ∨, ∧ and \ holds for all skew Boolean algebras if
and only if it holds on 5. It holds for all left-handed (right-handed) skew Boolean algebras if and
only if it holds on 3L (or 3R). The question of when a (quasi-)identity holds in any of these
varieties is thus decidable. £
The above results reveal the following simple lattice. Here 〈A〉 denotes the subvariety
generated by the algebra A.
123
