Page 272 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Ternary discriminators and ternary discriminator varieties
A ternary discriminator on a set S is a function d: S3 → S defined on a given set S by
d(x, y, z) = ⎧x if x ≠ y .
⎨⎩z otherwise
An algebra A is a ternary discriminator algebra if d can be polynomial-defined on its
underlying set A. Ternary discriminator algebras are simple. Indeed given elements a ≠ b in
such an algebra A and let θ be a congruence on A for which a θ b. Then for all c ∈ A,
a = d(a, b, c) θ d(a, a, c) = c.
Thus the only non-identity congruence on A is the universal congruence. Murskii [1975] proved
that in a certain sense almost all finite algebras are ternary discriminator algebras.
If K is a class of algebras of common type having a common ternary discriminator term,
the variety V generated from K is a ternary discriminator variety. Burris and Sankappanavar
[1981] described such a variety as “the most successful generalization of Boolean algebras to
date, successful because we obtain Boolean product representations.” Examples of these varieties
include Boolean algebras, n-dimensional cylindric algebras, p-rings and skew Boolean ∩-
algebras. The remaining results in this section are from Bignall and Leech [1995].
Theorem 7.1.10. In the variety of skew Boolean ∩-algebras the polynomial term
d(x, y, z) = (x / y) ∨ [z \ ( (x / y) ∨ (y / x) )]
is a ternary discriminator on any primitive algebra. Thus skew Boolean ∩-algebras form a
ternary discriminator variety. As such they are both congruence distributive and congruence
permutable. (The congruence distributive property was already observed in Theorem 4.4.3.)
Proof. If x = y, then clearly d(x, y, z) = z on any algebra, primitive or otherwise. If x ≠ y on
some primitive algebra P, then x∩y = 0, so that x / y = x and y / x = y on P. Thus the displayed
polynomial reduces to x ∨ [z \ (x ∨ y)]. If x ≠ 0, the latter reduces to x ∨ 0 = x. If x = 0, then y ≠
0 so that x ∨ [z \ (x ∨ y)] reduces to z \ y = 0 = x on P. The congruence distributive property has
already been seen in the case of ∩-algebras. That it and the congruence permutable property hold
on all discriminator varieties follows from results of Bulman-Flemming, Keimel and Werner.
(See Theorem IV.9.4 in Burris and Sankppanavar [1981].) £
A pointed ternary discriminator variety is a ternary discriminator variety with a constant
term 0. PD0 denotes the pointed ternary discriminator variety generated by the class of all
ternary discriminator algebras (A; d, 0) with 0 as a nullary operation, i.e., a constant.
270
Ternary discriminators and ternary discriminator varieties
A ternary discriminator on a set S is a function d: S3 → S defined on a given set S by
d(x, y, z) = ⎧x if x ≠ y .
⎨⎩z otherwise
An algebra A is a ternary discriminator algebra if d can be polynomial-defined on its
underlying set A. Ternary discriminator algebras are simple. Indeed given elements a ≠ b in
such an algebra A and let θ be a congruence on A for which a θ b. Then for all c ∈ A,
a = d(a, b, c) θ d(a, a, c) = c.
Thus the only non-identity congruence on A is the universal congruence. Murskii [1975] proved
that in a certain sense almost all finite algebras are ternary discriminator algebras.
If K is a class of algebras of common type having a common ternary discriminator term,
the variety V generated from K is a ternary discriminator variety. Burris and Sankappanavar
[1981] described such a variety as “the most successful generalization of Boolean algebras to
date, successful because we obtain Boolean product representations.” Examples of these varieties
include Boolean algebras, n-dimensional cylindric algebras, p-rings and skew Boolean ∩-
algebras. The remaining results in this section are from Bignall and Leech [1995].
Theorem 7.1.10. In the variety of skew Boolean ∩-algebras the polynomial term
d(x, y, z) = (x / y) ∨ [z \ ( (x / y) ∨ (y / x) )]
is a ternary discriminator on any primitive algebra. Thus skew Boolean ∩-algebras form a
ternary discriminator variety. As such they are both congruence distributive and congruence
permutable. (The congruence distributive property was already observed in Theorem 4.4.3.)
Proof. If x = y, then clearly d(x, y, z) = z on any algebra, primitive or otherwise. If x ≠ y on
some primitive algebra P, then x∩y = 0, so that x / y = x and y / x = y on P. Thus the displayed
polynomial reduces to x ∨ [z \ (x ∨ y)]. If x ≠ 0, the latter reduces to x ∨ 0 = x. If x = 0, then y ≠
0 so that x ∨ [z \ (x ∨ y)] reduces to z \ y = 0 = x on P. The congruence distributive property has
already been seen in the case of ∩-algebras. That it and the congruence permutable property hold
on all discriminator varieties follows from results of Bulman-Flemming, Keimel and Werner.
(See Theorem IV.9.4 in Burris and Sankppanavar [1981].) £
A pointed ternary discriminator variety is a ternary discriminator variety with a constant
term 0. PD0 denotes the pointed ternary discriminator variety generated by the class of all
ternary discriminator algebras (A; d, 0) with 0 as a nullary operation, i.e., a constant.
270
