Page 267 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 267
: Further Topics in Skew Boolean Algebras
These definitions generalize the concepts of pointed ternary discriminator and pointed
ternary discriminator variety. Indeed from a ternary discriminator d(x, y, z) and 0 one defines the
binary discriminator by setting x \ y = d(0, y, x). Thus pointed ternary discriminator varieties are
binary discriminator varieties and all the later are dual binary discriminator varieties. These
relationships cannot be reversed in general. Allowing for a bit of repetition we have:
Theorem 7.1.4. Skew Boolean algebras form a binary discriminator variety with \ being
a binary 0-discriminator on primitive skew Boolean algebras. Thus if (S: ∨, ∧, \, 0) is a skew
Boolean algebra, then the reduct (S: \, 0) is an iBCS algebra. £
It is well known that the variety of left normal bands with zero is generated by the three-
element band ({0, 1, 2}; ∧, 0), where ∧ is the dual binary discriminator on the base set {0, 1, 2}.
Since any algebra of the form (A; ∧, 0) is a left normal band when ∧ is the dual binary
discriminator on A, the class of left normal bands with zero is called the generic dual binary
discriminator variety. Strongly distributive skew lattices with zero provide another example of a
dual binary discriminator variety.
Before giving a semigroup characterization of iBCS-algebras we will need some further
properties of iBCS-algebras. We follow Matthew Spinks’ Monash University dissertation [2002].
Lemma 7.1.5. Upon setting x∧y = x \ (x \ y), an iBCS-algebra also satisfies identities:
B13. (a \ b) \ (c \ a) ≈ a \ b. B14. a \ (b \ (c \ a)) ≈ a \ b.
B15. a \ (a \ (a \ b)) ≈ a \ b. B16. (a ∧ b) \ c = (a \ c) \ (a \ b).
B17. (a \ c) \ b = (a \ c) \ (a \ b). B18. (a \ c) ∧ (b \ c) = (a \ c) \ (a \ b).
B19. a ∧ (b \ c) = (a \ c) \ (a \ b).
Proof.
B13. (a \ b) \ (c \ a) =B7 [a \ (c \ a)] \ b =B9 a \ b.
B14. a \ (b \ (c \ a)) =B9 [a \ (c \ a)] \ [b \ (c \ a)] =B8 (a \ b) \ (c \ a) =B13 a \ b.
B15. a \ b =B9 (a \ b) \ [a \ (a \ b)] =B8 {a \ [a \ (a \ b)]} \ {b \ [a \ (a \ b)]}
=B13 {a \ [a \ (a \ b)]} \ (b \ a) =B7 [a \ (b \ a)] \ [a \ (a \ b)]
=B9 a \ (a \ (a \ b)).
B16. (a ∧ b) \ c =B5 [a \ (a \ b)] \ c =B7 (a \ c) \ (a \ b).
B17. (a \ c) ∧ b =B5 (a \ c) \ [(a \ c) \ b] =B7 (a \ c) \ [(a \ b) \ c]
=B8 [a \ (a \ b)] \ c =B5 (a ∧ b) \ c =B16 (a \ c) \ (a \ b).
B18. (a \ c) ∧ (b \ c) =B5 (a \ c) \ [(a \ c) \ (b \ c)] =B8 (a \ c) \ [(a \ b) \ c]
=B7 (a \ c) \ [((a \ c) \ b] =B5 (a \ c) ∧ b =B17 (a \ c) \ (a \ b).
B19. a ∧ (b \ c) =B5 a \ [(a \ (b \ c)] =B9 [a \ (c \ a)] \ [a \ (b \ c)]
=B7 (a \ [a \ (b \ c)]) \ (c \ a)
265
These definitions generalize the concepts of pointed ternary discriminator and pointed
ternary discriminator variety. Indeed from a ternary discriminator d(x, y, z) and 0 one defines the
binary discriminator by setting x \ y = d(0, y, x). Thus pointed ternary discriminator varieties are
binary discriminator varieties and all the later are dual binary discriminator varieties. These
relationships cannot be reversed in general. Allowing for a bit of repetition we have:
Theorem 7.1.4. Skew Boolean algebras form a binary discriminator variety with \ being
a binary 0-discriminator on primitive skew Boolean algebras. Thus if (S: ∨, ∧, \, 0) is a skew
Boolean algebra, then the reduct (S: \, 0) is an iBCS algebra. £
It is well known that the variety of left normal bands with zero is generated by the three-
element band ({0, 1, 2}; ∧, 0), where ∧ is the dual binary discriminator on the base set {0, 1, 2}.
Since any algebra of the form (A; ∧, 0) is a left normal band when ∧ is the dual binary
discriminator on A, the class of left normal bands with zero is called the generic dual binary
discriminator variety. Strongly distributive skew lattices with zero provide another example of a
dual binary discriminator variety.
Before giving a semigroup characterization of iBCS-algebras we will need some further
properties of iBCS-algebras. We follow Matthew Spinks’ Monash University dissertation [2002].
Lemma 7.1.5. Upon setting x∧y = x \ (x \ y), an iBCS-algebra also satisfies identities:
B13. (a \ b) \ (c \ a) ≈ a \ b. B14. a \ (b \ (c \ a)) ≈ a \ b.
B15. a \ (a \ (a \ b)) ≈ a \ b. B16. (a ∧ b) \ c = (a \ c) \ (a \ b).
B17. (a \ c) \ b = (a \ c) \ (a \ b). B18. (a \ c) ∧ (b \ c) = (a \ c) \ (a \ b).
B19. a ∧ (b \ c) = (a \ c) \ (a \ b).
Proof.
B13. (a \ b) \ (c \ a) =B7 [a \ (c \ a)] \ b =B9 a \ b.
B14. a \ (b \ (c \ a)) =B9 [a \ (c \ a)] \ [b \ (c \ a)] =B8 (a \ b) \ (c \ a) =B13 a \ b.
B15. a \ b =B9 (a \ b) \ [a \ (a \ b)] =B8 {a \ [a \ (a \ b)]} \ {b \ [a \ (a \ b)]}
=B13 {a \ [a \ (a \ b)]} \ (b \ a) =B7 [a \ (b \ a)] \ [a \ (a \ b)]
=B9 a \ (a \ (a \ b)).
B16. (a ∧ b) \ c =B5 [a \ (a \ b)] \ c =B7 (a \ c) \ (a \ b).
B17. (a \ c) ∧ b =B5 (a \ c) \ [(a \ c) \ b] =B7 (a \ c) \ [(a \ b) \ c]
=B8 [a \ (a \ b)] \ c =B5 (a ∧ b) \ c =B16 (a \ c) \ (a \ b).
B18. (a \ c) ∧ (b \ c) =B5 (a \ c) \ [(a \ c) \ (b \ c)] =B8 (a \ c) \ [(a \ b) \ c]
=B7 (a \ c) \ [((a \ c) \ b] =B5 (a \ c) ∧ b =B17 (a \ c) \ (a \ b).
B19. a ∧ (b \ c) =B5 a \ [(a \ (b \ c)] =B9 [a \ (c \ a)] \ [a \ (b \ c)]
=B7 (a \ [a \ (b \ c)]) \ (c \ a)
265
