Page 266 - 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
(Dual) binary discriminator varieties and iBCS algebras
Let A be a set with distinguished element 0. The binary 0-discriminator and the dual
binary 0-discriminator on A are defined respectively by
x \ y = ⎧⎩⎨0x ioft hye r=w i0se and x∧y = ⎨⎧⎩0x ioft hye r=w i0se .
Clearly x \ y and x∧y are left-handed skew Boolean operations on A as a primitive skew Boolean
algebra. When no ambiguity exists about the constant 0 we simply use the term (dual) binary
discriminator. The details of the following lemma are easily verified.
Lemma 7.1.3. Given a set A with constant 0, the functions x \ y and x∧y satisfy identities:
B1. a ∧ a = a. B2. a ∧ (b ∧ c) = (a ∧ b) ∧ c.
B3. a ∧ (b ∧ c) = a ∧ (c ∧ b). B4. a ∧ 0 = 0 ∧ a = 0.
B5. a ∧ b = a \ (a \ b). B6. a \ a = 0.
B7. (a \ b) \ c = (a \ c) \ b. B8. (a \ b) \ c = (a \ c) \ (b \ c).
B9. a \ (b \ a) = a. B10. a \ 0 = a.
B11. 0 \ a = a. B12. (a \ b) \ b = a \ b. £
Identities B1 to B4 characterize a left normal band with zero. Identities B6 through B9
reprise identities (a) – (d) at the onset of this section, with any algebra (A; \, 0) satisfying them
being an iBCS-algebra. For such algebras B10 – B12 also hold. B10 and B11 are just (e) and (f)
above, while
(a \ b) \ b =B8 (a \ b) \ (b \ b) =B6 (a \ b) \ 0 =B10 a \ b.
A variety V with a constant term 0 is a [dual] binary discriminator variety if a binary
term x \ y [x ∧ y] exits such that V is generated by a class K of algebras on which that binary term
induces the [dual] binary 0-discriminator.
Binary discriminator varieties are widespread. Examples include Stone algebras, pseudo-
complemented semilattices, implicative BCK-algebras and, as we will shortly show, any ternary
discriminator variety with a constant term.
Any binary discriminator variety is also a dual binary discriminator variety, due to B5,
but not conversely. Indeed the variety of left normal bands with zero is a dual binary 0-
discriminator variety, but it cannot be a binary 0-discriminator variety because any term t(x, y) in
the language of left normal bands with zero in which both variables appear explicitly must satisfy
the implication: if y = 0 then t(x, y) = 0. Hence a binary term x \ y satisfying the identity x \ 0 ≈ x
cannot be defined in the band.
264
(Dual) binary discriminator varieties and iBCS algebras
Let A be a set with distinguished element 0. The binary 0-discriminator and the dual
binary 0-discriminator on A are defined respectively by
x \ y = ⎧⎩⎨0x ioft hye r=w i0se and x∧y = ⎨⎧⎩0x ioft hye r=w i0se .
Clearly x \ y and x∧y are left-handed skew Boolean operations on A as a primitive skew Boolean
algebra. When no ambiguity exists about the constant 0 we simply use the term (dual) binary
discriminator. The details of the following lemma are easily verified.
Lemma 7.1.3. Given a set A with constant 0, the functions x \ y and x∧y satisfy identities:
B1. a ∧ a = a. B2. a ∧ (b ∧ c) = (a ∧ b) ∧ c.
B3. a ∧ (b ∧ c) = a ∧ (c ∧ b). B4. a ∧ 0 = 0 ∧ a = 0.
B5. a ∧ b = a \ (a \ b). B6. a \ a = 0.
B7. (a \ b) \ c = (a \ c) \ b. B8. (a \ b) \ c = (a \ c) \ (b \ c).
B9. a \ (b \ a) = a. B10. a \ 0 = a.
B11. 0 \ a = a. B12. (a \ b) \ b = a \ b. £
Identities B1 to B4 characterize a left normal band with zero. Identities B6 through B9
reprise identities (a) – (d) at the onset of this section, with any algebra (A; \, 0) satisfying them
being an iBCS-algebra. For such algebras B10 – B12 also hold. B10 and B11 are just (e) and (f)
above, while
(a \ b) \ b =B8 (a \ b) \ (b \ b) =B6 (a \ b) \ 0 =B10 a \ b.
A variety V with a constant term 0 is a [dual] binary discriminator variety if a binary
term x \ y [x ∧ y] exits such that V is generated by a class K of algebras on which that binary term
induces the [dual] binary 0-discriminator.
Binary discriminator varieties are widespread. Examples include Stone algebras, pseudo-
complemented semilattices, implicative BCK-algebras and, as we will shortly show, any ternary
discriminator variety with a constant term.
Any binary discriminator variety is also a dual binary discriminator variety, due to B5,
but not conversely. Indeed the variety of left normal bands with zero is a dual binary 0-
discriminator variety, but it cannot be a binary 0-discriminator variety because any term t(x, y) in
the language of left normal bands with zero in which both variables appear explicitly must satisfy
the implication: if y = 0 then t(x, y) = 0. Hence a binary term x \ y satisfying the identity x \ 0 ≈ x
cannot be defined in the band.
264
