Page 127 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 127
IV: Skew Boolean Algebras
directly above 0. An atom or primitive element is any element in a primitive class. A skew
Boolean algebra is atomic if for every nonzero x ∈S an atom a ∈S exists such that a ≤ x.
Theorem 4.1.12. A skew Boolean algebra is completely reducible if and only if it is
complete and atomic.
Proof. Completely reducible algebras are clearly complete and atomic. For the converse, let S
be a complete atomic skew Boolean algebra. For any x ∈S, the set x∧S∧x is a complete atomic
Boolean algebra with (unique) maximal element x. Thus each x > 0 in S is the supremum of its
underlying atoms, all of which commute, with distinct sets of commuting atoms giving distinct
suprema. Next, let {Xj} be an indexed collection of all primitive classes of S and let Pj = Xj0
be the corresponding primitive skew Boolean algebras. Define σ: ∏j Pj → S by setting
σ[〈ej〉] = sup〈ej〉. By our remarks, σ is at least a bijection.
Let 〈ej〉, 〈fj〉 ∈ ∏j Pj be given with supj〈ej〉 = e and supj〈fj〉 = f. Then e∨f ≥ ej∨fj for each j.
Indeed,
(e∨f) ∧ (ej∨fj) = (e∧ej) ∨ (f∧ej) ∨ (e∧fj) ∨ (f∧fj) = ej ∨ (f∧ej) ∨ (e∧fj) ∨ fj = ej ∨ (f∧ej) ∨ fj.
Case 1. Both ej, fj ≠ 0. Here ej∨(f∧ej)∨fj = ej∨fj since all ∨-factors are D-equivalent.
Case 2. fj = 0. Then f∧ej = 0 since its image in S/D is 0. Hence fj∨(f∧ej)∨fj = ej = ej∨fj.
Case 3. ej = 0. Here ej∨(f∧ej)∨fj = fj = ej∨fj again.
Similarly, (ej∨fj)∧(e∨f) = (ej∨fj) so that e∨f ≥ ej∨fj is verified. Each primitive join ej∨fj is either
an atom of S or 0. Claim e∨f = sup(ej∨fj). Clearly e∨f ≥ sup(ej∨fj). If the inequality is strict, this
means that e∨f ≥ an atom gk that does not lie below sup(ej∨fj). Thus for this index k, ek = 0 = fk.
But this yields a contradiction in the Boolean lattice S/D where Dgk is an atom and neither De ≥
Dgk nor Df ≥ Dgk , yet De ∨ Df ≥ Dgk . Thus e∨y = sup(ej∨fj) as claimed. Next, by mid-
commutativity, (e∧f) ∧ (ej∧fj) = (e∧ej) ∧ (f∧fj) = ej∧fj, and similarly (ej∧fj) ∧ (e∧f) = ej∧fj. Thus
(e∧f) ≥ (ej∧fj) for each j and hence (e∧f) ≥ sup(ej∧fj). An argument similar to that for the join
guarantees that (e∧f) = sup(ej∧fj). Thus σ: ∏j Pj → S is an isomorphism of skew lattices. Since \
is implicitly determined by the skew lattice structure, σ is an isomorphism and S is seen to be
completely reducible. £
Theorem 4.1.13. Complete skew Boolean algebras satisfy the identities:
1) e ∧ sup(fi) = sup(e ∧ fi).
2) e ∨ inf(fi) = inf(e ∨ fi).
3) e \ sup(fi) = inf(e \ fi).
4) e \ inf(fi) = sup(e \ fi).
Proof. In each case the obvious inequality (≤ or ≥) becomes equality in its complete maximal
Boolean algebra image. Thus one already has equality in the skew Boolean algebra. £
125
directly above 0. An atom or primitive element is any element in a primitive class. A skew
Boolean algebra is atomic if for every nonzero x ∈S an atom a ∈S exists such that a ≤ x.
Theorem 4.1.12. A skew Boolean algebra is completely reducible if and only if it is
complete and atomic.
Proof. Completely reducible algebras are clearly complete and atomic. For the converse, let S
be a complete atomic skew Boolean algebra. For any x ∈S, the set x∧S∧x is a complete atomic
Boolean algebra with (unique) maximal element x. Thus each x > 0 in S is the supremum of its
underlying atoms, all of which commute, with distinct sets of commuting atoms giving distinct
suprema. Next, let {Xj} be an indexed collection of all primitive classes of S and let Pj = Xj0
be the corresponding primitive skew Boolean algebras. Define σ: ∏j Pj → S by setting
σ[〈ej〉] = sup〈ej〉. By our remarks, σ is at least a bijection.
Let 〈ej〉, 〈fj〉 ∈ ∏j Pj be given with supj〈ej〉 = e and supj〈fj〉 = f. Then e∨f ≥ ej∨fj for each j.
Indeed,
(e∨f) ∧ (ej∨fj) = (e∧ej) ∨ (f∧ej) ∨ (e∧fj) ∨ (f∧fj) = ej ∨ (f∧ej) ∨ (e∧fj) ∨ fj = ej ∨ (f∧ej) ∨ fj.
Case 1. Both ej, fj ≠ 0. Here ej∨(f∧ej)∨fj = ej∨fj since all ∨-factors are D-equivalent.
Case 2. fj = 0. Then f∧ej = 0 since its image in S/D is 0. Hence fj∨(f∧ej)∨fj = ej = ej∨fj.
Case 3. ej = 0. Here ej∨(f∧ej)∨fj = fj = ej∨fj again.
Similarly, (ej∨fj)∧(e∨f) = (ej∨fj) so that e∨f ≥ ej∨fj is verified. Each primitive join ej∨fj is either
an atom of S or 0. Claim e∨f = sup(ej∨fj). Clearly e∨f ≥ sup(ej∨fj). If the inequality is strict, this
means that e∨f ≥ an atom gk that does not lie below sup(ej∨fj). Thus for this index k, ek = 0 = fk.
But this yields a contradiction in the Boolean lattice S/D where Dgk is an atom and neither De ≥
Dgk nor Df ≥ Dgk , yet De ∨ Df ≥ Dgk . Thus e∨y = sup(ej∨fj) as claimed. Next, by mid-
commutativity, (e∧f) ∧ (ej∧fj) = (e∧ej) ∧ (f∧fj) = ej∧fj, and similarly (ej∧fj) ∧ (e∧f) = ej∧fj. Thus
(e∧f) ≥ (ej∧fj) for each j and hence (e∧f) ≥ sup(ej∧fj). An argument similar to that for the join
guarantees that (e∧f) = sup(ej∧fj). Thus σ: ∏j Pj → S is an isomorphism of skew lattices. Since \
is implicitly determined by the skew lattice structure, σ is an isomorphism and S is seen to be
completely reducible. £
Theorem 4.1.13. Complete skew Boolean algebras satisfy the identities:
1) e ∧ sup(fi) = sup(e ∧ fi).
2) e ∨ inf(fi) = inf(e ∨ fi).
3) e \ sup(fi) = inf(e \ fi).
4) e \ inf(fi) = sup(e \ fi).
Proof. In each case the obvious inequality (≤ or ≥) becomes equality in its complete maximal
Boolean algebra image. Thus one already has equality in the skew Boolean algebra. £
125
