Page 151 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 151
IV: Skew Boolean Algebras
Proof. Clearly we have a diagram of inclusions. Since any equation in the operations of the
signature contains only finitely many variables, the final three assertions are also clear. Thus we
need only show the inclusions of the induced subvarieties to be proper in the case of the finite
primitive algebras. We first show 〈nL〉∩ ⊂ 〈n+1L〉∩ to be proper for all finite n. To begin, x ≈ y
holds on 〈1〉∩ but not on 〈2〉∩. Next, for n ≥ 2 we set
Φn(x1, x2, . . . , xn) = (x1 \ (x1∩x2)) ∧ (x2 \ (x2∩x3)) ∧ . . . ∧ (xn \ (xn∩ x1))
and
Ψn(x1, x2, . . . , xn) = x1 \ [(x1∩x2) ∨ ... ∨ (x1∩xn) ∨ (x2∩x3) ∨ ... ∨ (x2∩xn) ∨ … ∨ (xn–1∩xn)].
Since Φn(a1, a2, . . . , an) ≠ 0 only if none of a1, a2, . . . , an is 0.
Φn(x1, x2, . . . , xn) ∧ Ψn(x1, x2, . . . , xn) ≈ 0
holds on nL but not on n+1L (respectively, on nR but not on n+1R) for all n ≥ 2. Thus all
inclusions at least along the two lower sides of the above diagram are proper. But this forces all
links in the above diagram to be proper. For instance, 〈mL•nR〉∩ ⊂ 〈m+1L•nR〉∩ for m < ℵ0 and
n ≤ ℵ0 is indeed proper since Φm(x1∧y, x2∧y, . . . , xm∧y) ∧ Ψm(x1∧y, x2∧y, . . . , xm∧y) ≈ 0 must
hold in 〈mL•nR〉∩ but not in 〈m+1L•nR〉∩. £
This ascending array of principal varieties leads us to several results, beginning with:
Theorem 4.4.20. Skew Boolean ∩-algebras are locally finite.
Proof. We revise the argument from Theorem 4.1.10. Given a skew Boolean ∩-algebra S
generated from a finite set X of size n, if ϕ: S → P is a nontrivial homomorphism from S to a
primitive algebra, then ϕ[S] is a primitive subalgebra Pʹ of P that is isomorphic to a subalgebra of
n+1L•n+1R. It follows that a homomorphism of ϕʹ: S → n+1L•n+1R exists inducing the same
congruence on S that ϕ has. Moreover, only finitely many distinct homomorphisms from S to
n+1L•n+1R are possible since S is generated from X. By the argument of Theorem 4.1.10, S can
be embedded in a finite power of n+1L•n+1R making S itself finite. £
Theorem 4.4.21. A (quasi-)identity of signature (∨, ∧, \, ∩, 0) in n variables holds for
all skew Boolean ∩-algebras if and only if it holds in n+1L•n+1R. Likewise the (quasi-)identity
holds for all left-handed (right-handed) skew Boolean ∩-algebras if and only if it holds in n+1L
(or in n+1R). The question of when a given (quasi-)identity holds for all (left-handed or right
handed) skew Boolean ∩-algebras is thus decidable. £
Proof. If a (quasi-)identity in variables x1, … , xn holds for all skew Boolean ∩-algebras, in
particular it holds for n+1L•n+1R. Conversely if it holds for n+1L•n+1R, it holds for all powers
of n+1L•n+1. Thus, given skew Boolean ∩-algebra S with a1, a2, …, an ∈ S, the latter
collectively generate a subalgebra Sʹ of S that (by previous arguments) can be embedded in some
149
Proof. Clearly we have a diagram of inclusions. Since any equation in the operations of the
signature contains only finitely many variables, the final three assertions are also clear. Thus we
need only show the inclusions of the induced subvarieties to be proper in the case of the finite
primitive algebras. We first show 〈nL〉∩ ⊂ 〈n+1L〉∩ to be proper for all finite n. To begin, x ≈ y
holds on 〈1〉∩ but not on 〈2〉∩. Next, for n ≥ 2 we set
Φn(x1, x2, . . . , xn) = (x1 \ (x1∩x2)) ∧ (x2 \ (x2∩x3)) ∧ . . . ∧ (xn \ (xn∩ x1))
and
Ψn(x1, x2, . . . , xn) = x1 \ [(x1∩x2) ∨ ... ∨ (x1∩xn) ∨ (x2∩x3) ∨ ... ∨ (x2∩xn) ∨ … ∨ (xn–1∩xn)].
Since Φn(a1, a2, . . . , an) ≠ 0 only if none of a1, a2, . . . , an is 0.
Φn(x1, x2, . . . , xn) ∧ Ψn(x1, x2, . . . , xn) ≈ 0
holds on nL but not on n+1L (respectively, on nR but not on n+1R) for all n ≥ 2. Thus all
inclusions at least along the two lower sides of the above diagram are proper. But this forces all
links in the above diagram to be proper. For instance, 〈mL•nR〉∩ ⊂ 〈m+1L•nR〉∩ for m < ℵ0 and
n ≤ ℵ0 is indeed proper since Φm(x1∧y, x2∧y, . . . , xm∧y) ∧ Ψm(x1∧y, x2∧y, . . . , xm∧y) ≈ 0 must
hold in 〈mL•nR〉∩ but not in 〈m+1L•nR〉∩. £
This ascending array of principal varieties leads us to several results, beginning with:
Theorem 4.4.20. Skew Boolean ∩-algebras are locally finite.
Proof. We revise the argument from Theorem 4.1.10. Given a skew Boolean ∩-algebra S
generated from a finite set X of size n, if ϕ: S → P is a nontrivial homomorphism from S to a
primitive algebra, then ϕ[S] is a primitive subalgebra Pʹ of P that is isomorphic to a subalgebra of
n+1L•n+1R. It follows that a homomorphism of ϕʹ: S → n+1L•n+1R exists inducing the same
congruence on S that ϕ has. Moreover, only finitely many distinct homomorphisms from S to
n+1L•n+1R are possible since S is generated from X. By the argument of Theorem 4.1.10, S can
be embedded in a finite power of n+1L•n+1R making S itself finite. £
Theorem 4.4.21. A (quasi-)identity of signature (∨, ∧, \, ∩, 0) in n variables holds for
all skew Boolean ∩-algebras if and only if it holds in n+1L•n+1R. Likewise the (quasi-)identity
holds for all left-handed (right-handed) skew Boolean ∩-algebras if and only if it holds in n+1L
(or in n+1R). The question of when a given (quasi-)identity holds for all (left-handed or right
handed) skew Boolean ∩-algebras is thus decidable. £
Proof. If a (quasi-)identity in variables x1, … , xn holds for all skew Boolean ∩-algebras, in
particular it holds for n+1L•n+1R. Conversely if it holds for n+1L•n+1R, it holds for all powers
of n+1L•n+1. Thus, given skew Boolean ∩-algebra S with a1, a2, …, an ∈ S, the latter
collectively generate a subalgebra Sʹ of S that (by previous arguments) can be embedded in some
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