Page 141 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 141
IV: Skew Boolean Algebras
Lemma 4.4.1. For any pair e, f in a skew lattice S the following are equivalent:
i) e∧f = f∧e.
ii) e∧f = e∩f, where e∩f exists.
iii) f∧e = e∩f, where e∩f exists.
Proof. That (i) implies (ii) and (iii) is clear. Conversely, given say (ii) one has
f ∧ e = f ∧ e ∧ f ∧ e = f ∧ (e∩f) ∧ e = e∩f = e ∧ f
and (i) follows. The equivalence of (i) and (iii) is similar. £
Lemma 4.4.2. A skew lattice having finite intersections is an algebra S = (S; ∨, ∧, ∩)
such that (S; ∩) is a meet semilattice, (S; ∨, ∧) is a skew lattice and the following identities hold:
e ∩ (e∧f∧e) = e∧f∧e and e ∧ ( e∩f) = e∩f = (e∩f) ∧ e.
Skew lattices with finite intersections thus form a variety of algebras.
Proof. The identities state in essence that the two partial orders on S induced by ∧ and ∩ must
contain each other and thus coincide. £
Theorem 4.4.3. (Theorem 1.3.11 restated) The variety of all skew lattices having finite
intersections is congruence distributive. £
A skew lattice (S; ∨, ∧) is initially finite if for all e ∈ S, ⎡e⎤ = {f ⎪f ≤ e} is finite. Clearly:
Theorem 4.4.4. If (S; ∨, ∧, 0) is a normal, symmetric skew lattice with a zero, then:
i) If S is join complete, then S is complete.
ii) If S is complete, then S has intersections.
iii) If S is initially finite, then S has intersections. £
A skew Boolean algebra with finite intersections is called a skew Boolean ∩-algebra.
(Read “skew Boolean intersection algebra”.) These algebras form a variety of algebras denoted
by SBA∩. We next scan examples of skew Boolean algebras to see which are ∩-algebras.
Example 4.4.1. Finite intersections trivially exist for generalized Boolean algebras.
Example 4.4.2. A 0-primitive skew Boolean algebra S = D0 has arbitrary intersections.
Given A ⊆ S, infA is the sole element of A when A is a singleton set, and 0 otherwise.
Example 4.4.3. Every completely reducible skew Boolean algebra, being the product of
primitive algebras, has arbitrary intersections.
139
Lemma 4.4.1. For any pair e, f in a skew lattice S the following are equivalent:
i) e∧f = f∧e.
ii) e∧f = e∩f, where e∩f exists.
iii) f∧e = e∩f, where e∩f exists.
Proof. That (i) implies (ii) and (iii) is clear. Conversely, given say (ii) one has
f ∧ e = f ∧ e ∧ f ∧ e = f ∧ (e∩f) ∧ e = e∩f = e ∧ f
and (i) follows. The equivalence of (i) and (iii) is similar. £
Lemma 4.4.2. A skew lattice having finite intersections is an algebra S = (S; ∨, ∧, ∩)
such that (S; ∩) is a meet semilattice, (S; ∨, ∧) is a skew lattice and the following identities hold:
e ∩ (e∧f∧e) = e∧f∧e and e ∧ ( e∩f) = e∩f = (e∩f) ∧ e.
Skew lattices with finite intersections thus form a variety of algebras.
Proof. The identities state in essence that the two partial orders on S induced by ∧ and ∩ must
contain each other and thus coincide. £
Theorem 4.4.3. (Theorem 1.3.11 restated) The variety of all skew lattices having finite
intersections is congruence distributive. £
A skew lattice (S; ∨, ∧) is initially finite if for all e ∈ S, ⎡e⎤ = {f ⎪f ≤ e} is finite. Clearly:
Theorem 4.4.4. If (S; ∨, ∧, 0) is a normal, symmetric skew lattice with a zero, then:
i) If S is join complete, then S is complete.
ii) If S is complete, then S has intersections.
iii) If S is initially finite, then S has intersections. £
A skew Boolean algebra with finite intersections is called a skew Boolean ∩-algebra.
(Read “skew Boolean intersection algebra”.) These algebras form a variety of algebras denoted
by SBA∩. We next scan examples of skew Boolean algebras to see which are ∩-algebras.
Example 4.4.1. Finite intersections trivially exist for generalized Boolean algebras.
Example 4.4.2. A 0-primitive skew Boolean algebra S = D0 has arbitrary intersections.
Given A ⊆ S, infA is the sole element of A when A is a singleton set, and 0 otherwise.
Example 4.4.3. Every completely reducible skew Boolean algebra, being the product of
primitive algebras, has arbitrary intersections.
139
