Page 157 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 157
IV: Skew Boolean Algebras
Since ∧ is a left normal operation, e1∧e2 ∩ (f1∧e2) = e1 ∩ e2 ∩ (f1∧e2) = e1 ∩ f1 ∩ e2. Indeed we
have e1 ∩ e2 ∩ (f1∧e2) ≤ e1 ∩ e2 ∩ f1 = e1 ∩ f1 ∩ e2. But on the other hand, f1 ∩ e2 ≤ f1∧e2 so that
e1 ∩ f1 ∩ e2 ≤ e1 ∩ (f1∧e2) ∩ e2 = e1 ∩ e2 ∩ (f1∧e2). β also preserves ∨. Given e1 and e2 in S0, then
β[f1∨f2] = (e1∨e2, (e1∨e2) ∩ (f1∨f2)) and
β[f1]∨β[f2] = (e1, e1 ∩ f1) ∨ (e2, e2 ∩ f2) = (e1∨e2, [(e1 ∩ f1) \ e2] ∨ [e2 ∩ f2])
for all f1 D e1 and f2 D e2 in S. It remains to show that
(e1∨e2) ∩ (f1∨f2) = ((e1 ∩ f1) \ e2) ∨ (e2 ∩ f2)
is an identity for left-handed skew Boolean ∩-algebras, subject to the conditions f1 D e1 and
f2 D e2. This is indeed the case for left-handed primitive skew Boolean ∩-algebras. To see this,
first assume e1 = f1 = 0. Here the equation reduces to the identity e2 ∩ f2 = 0 ∨ (e2 ∩ f2).
Likewise, in the case e2 = f2 = 0 the equation reduces to the identity e1 ∩ f1 = ((e1 ∩ f1) \ 0) ∨ 0.
Otherwise all four elements lie in the unique nonzero class and we get e2 ∩ f2 = 0 ∨ (e2 ∩ f2).
Thus the conditional equality holds on primitive algebras. But this explicit D-condition can be
removed upon replacing e1, f1, e2 and f2 by e1∧f1, f1∧e1, e2∧f2 and f2∧e2 respectively. Hence the
above conditional identity holds for all left-handed skew Boolean ∩-algebras. We have shown
that β also preserves joins and hence β: S ≅ ω(S0). £
Example 4.5.1. In particular, a primitive skew Boolean algebra S is isomorphic to an
omega algebra if and only if it is left-handed of order 3. In this case we are looking at a copy of
3L, which is what ω(2) is for the primitive Boolean algebra 2: 1 > 0.
1– 2 ω(2): (1,1) – (1, 0)
3L: . .
Theorem 4.5.9. ω(B) is complete if and only if B is complete. It is both complete and
atomic if and only if B is thus, in which case ω(B) ≅ 3L |A(B)| where A(B) is the set of atoms of B.
Thus a complete, atomic skew Boolean algebra is isomorphic to some omega algebra if and only
if it is left-handed and each of its atomic D-classes has exactly two elements.
Proof. If B is complete and A = {(ai, aʹi) ⎢i ∈ I} is a pairwise commuting subset of ω(B), we
claim that sup(A) = (supaj, supaʹj) with both suprema taken in B. By Lemma 4.5.4, if
(b, bʹ) ≥ (ai, aʹi), then b ≥ supaj and bʹ ≥ supaʹj. Thus, we need only show (supaj, supaʹj) ≥ each
(ai, aʹi) in ω(B). Lemma 4.5.3 and the definition of ∧ in ω(B) give
ai ∧ supj aʹj = supj (ai ∧ aʹj) = supj (aʹi ∧ aʹj) = aʹi
which is what Lemma 4.5.4 requires for (supaj, supaʹj) ≥ (ai, aʹi). Since ω(B) is normal with 0
and has commuting suprema, all nonempty subsets of ω(B) also have infima.
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Since ∧ is a left normal operation, e1∧e2 ∩ (f1∧e2) = e1 ∩ e2 ∩ (f1∧e2) = e1 ∩ f1 ∩ e2. Indeed we
have e1 ∩ e2 ∩ (f1∧e2) ≤ e1 ∩ e2 ∩ f1 = e1 ∩ f1 ∩ e2. But on the other hand, f1 ∩ e2 ≤ f1∧e2 so that
e1 ∩ f1 ∩ e2 ≤ e1 ∩ (f1∧e2) ∩ e2 = e1 ∩ e2 ∩ (f1∧e2). β also preserves ∨. Given e1 and e2 in S0, then
β[f1∨f2] = (e1∨e2, (e1∨e2) ∩ (f1∨f2)) and
β[f1]∨β[f2] = (e1, e1 ∩ f1) ∨ (e2, e2 ∩ f2) = (e1∨e2, [(e1 ∩ f1) \ e2] ∨ [e2 ∩ f2])
for all f1 D e1 and f2 D e2 in S. It remains to show that
(e1∨e2) ∩ (f1∨f2) = ((e1 ∩ f1) \ e2) ∨ (e2 ∩ f2)
is an identity for left-handed skew Boolean ∩-algebras, subject to the conditions f1 D e1 and
f2 D e2. This is indeed the case for left-handed primitive skew Boolean ∩-algebras. To see this,
first assume e1 = f1 = 0. Here the equation reduces to the identity e2 ∩ f2 = 0 ∨ (e2 ∩ f2).
Likewise, in the case e2 = f2 = 0 the equation reduces to the identity e1 ∩ f1 = ((e1 ∩ f1) \ 0) ∨ 0.
Otherwise all four elements lie in the unique nonzero class and we get e2 ∩ f2 = 0 ∨ (e2 ∩ f2).
Thus the conditional equality holds on primitive algebras. But this explicit D-condition can be
removed upon replacing e1, f1, e2 and f2 by e1∧f1, f1∧e1, e2∧f2 and f2∧e2 respectively. Hence the
above conditional identity holds for all left-handed skew Boolean ∩-algebras. We have shown
that β also preserves joins and hence β: S ≅ ω(S0). £
Example 4.5.1. In particular, a primitive skew Boolean algebra S is isomorphic to an
omega algebra if and only if it is left-handed of order 3. In this case we are looking at a copy of
3L, which is what ω(2) is for the primitive Boolean algebra 2: 1 > 0.
1– 2 ω(2): (1,1) – (1, 0)
3L: . .
Theorem 4.5.9. ω(B) is complete if and only if B is complete. It is both complete and
atomic if and only if B is thus, in which case ω(B) ≅ 3L |A(B)| where A(B) is the set of atoms of B.
Thus a complete, atomic skew Boolean algebra is isomorphic to some omega algebra if and only
if it is left-handed and each of its atomic D-classes has exactly two elements.
Proof. If B is complete and A = {(ai, aʹi) ⎢i ∈ I} is a pairwise commuting subset of ω(B), we
claim that sup(A) = (supaj, supaʹj) with both suprema taken in B. By Lemma 4.5.4, if
(b, bʹ) ≥ (ai, aʹi), then b ≥ supaj and bʹ ≥ supaʹj. Thus, we need only show (supaj, supaʹj) ≥ each
(ai, aʹi) in ω(B). Lemma 4.5.3 and the definition of ∧ in ω(B) give
ai ∧ supj aʹj = supj (ai ∧ aʹj) = supj (aʹi ∧ aʹj) = aʹi
which is what Lemma 4.5.4 requires for (supaj, supaʹj) ≥ (ai, aʹi). Since ω(B) is normal with 0
and has commuting suprema, all nonempty subsets of ω(B) also have infima.
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