Page 271 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 271
0: VII: Further Topics in Skew Boolean Algebras
b = 0.
L = (c + (b \ c)) + (0 \ (c + (b \ c)) = (c + (b \ c)) + 0 = c + (b \ c).
R = c + ((b + (0 \ b)) \ c) = c + ((b + 0) \ c) = c + (b \ c) = L.
L = (c + 0) + ((a \ (c + 0)) = c + (a \ c).
R = c + (0 + (a \ 0) \ c) = c + ((0 + a) \ c) = c + (a \ c) = L
c = 0: In a similar fashion, L = b + (a \ b) = R.
a, b, c ≠ 0. Thus a \ b = b \ c = a \ c = 0 so that
L = (c + 0) + (a \ (c + 0)) = c + (a \ c) = c + 0 = c, while
R = c + ((b + 0) \ c) = c + (b \ c) = c + 0 = c.
Thus a ∨ (b ∨ c) = (a ∨ b) ∨ c in all possible cases. The absorption identities are similarly
checked for \ being the binary discriminator. We consider only a ∧ (a ∨ b) = a. Here there are
three cases.
a = 0: 0 ∧ (0 ∨ b) = 0 ∧ (b + 0\b) = 0 ∧ (b + 0) = 0 ∧ b = 0.
b = 0: a ∧ (a ∨ 0) = a ∧ (0 + a\0) = a ∧ (0 + a) = a ∧ a = a.
a, b ≠ 0: a ∧ (a ∨ b) = a ∧ (b + a\b) = a ∧ (b + 0) = a ∧ b = a.
Thus (S; ∨, ∧, 0) is at least a normal skew lattice with a zero. To show symmetry we verify that
a∧b = b∧a implies a∨b = b∨a. (The converse holds for all normal skew lattices.) Since S is
primitive, two main (nonexclusive) cases of a∧b = b∧a occur. Either a or b is 0, say a = 0. Here
while 0 ∨ b = b + (0 \ b) = b
b ∨ 0 = 0 + (b \ 0) = 0 + b = b.
The other case is a equals b. Here commutativity of ∨ is trivial. Thus (S; ∨, ∧) is symmetric,
normal skew lattice. The rest follows from Theorem 7.1.6. £
Example 7.1.1. Let SA denote the variety of Stone algebras. A member of SA has the
form (A; ∩, ∪, *, 0,1), where (A; ∩, ∪, 0,1) is a bounded distributive lattice and * is a relative
pseudo-complementation operation; thus a* is the largest element of A such that a ∩ a* = 0.
Stone algebras form a subvariety of the variety of pseudo-complemented distributive lattices.
They are distinguished from other pseudo-complemented distributive lattices by the identity x* ≈
x***. Since they are generated by the two and three-element chains, it is not difficult to see that
Stone algebras form a binary discriminator variety with binary discriminator term x \ y = x ∩ y*.
For a, b ∈ A let a ∨ b be b ∪ (a ∩ b*) and a ∧ b be a ∩ (a ∩ b*)*. Then by theorem above, the
derived algebra (A; ∨, ∧, \, 0) is a left-handed skew Boolean algebra. £
269
b = 0.
L = (c + (b \ c)) + (0 \ (c + (b \ c)) = (c + (b \ c)) + 0 = c + (b \ c).
R = c + ((b + (0 \ b)) \ c) = c + ((b + 0) \ c) = c + (b \ c) = L.
L = (c + 0) + ((a \ (c + 0)) = c + (a \ c).
R = c + (0 + (a \ 0) \ c) = c + ((0 + a) \ c) = c + (a \ c) = L
c = 0: In a similar fashion, L = b + (a \ b) = R.
a, b, c ≠ 0. Thus a \ b = b \ c = a \ c = 0 so that
L = (c + 0) + (a \ (c + 0)) = c + (a \ c) = c + 0 = c, while
R = c + ((b + 0) \ c) = c + (b \ c) = c + 0 = c.
Thus a ∨ (b ∨ c) = (a ∨ b) ∨ c in all possible cases. The absorption identities are similarly
checked for \ being the binary discriminator. We consider only a ∧ (a ∨ b) = a. Here there are
three cases.
a = 0: 0 ∧ (0 ∨ b) = 0 ∧ (b + 0\b) = 0 ∧ (b + 0) = 0 ∧ b = 0.
b = 0: a ∧ (a ∨ 0) = a ∧ (0 + a\0) = a ∧ (0 + a) = a ∧ a = a.
a, b ≠ 0: a ∧ (a ∨ b) = a ∧ (b + a\b) = a ∧ (b + 0) = a ∧ b = a.
Thus (S; ∨, ∧, 0) is at least a normal skew lattice with a zero. To show symmetry we verify that
a∧b = b∧a implies a∨b = b∨a. (The converse holds for all normal skew lattices.) Since S is
primitive, two main (nonexclusive) cases of a∧b = b∧a occur. Either a or b is 0, say a = 0. Here
while 0 ∨ b = b + (0 \ b) = b
b ∨ 0 = 0 + (b \ 0) = 0 + b = b.
The other case is a equals b. Here commutativity of ∨ is trivial. Thus (S; ∨, ∧) is symmetric,
normal skew lattice. The rest follows from Theorem 7.1.6. £
Example 7.1.1. Let SA denote the variety of Stone algebras. A member of SA has the
form (A; ∩, ∪, *, 0,1), where (A; ∩, ∪, 0,1) is a bounded distributive lattice and * is a relative
pseudo-complementation operation; thus a* is the largest element of A such that a ∩ a* = 0.
Stone algebras form a subvariety of the variety of pseudo-complemented distributive lattices.
They are distinguished from other pseudo-complemented distributive lattices by the identity x* ≈
x***. Since they are generated by the two and three-element chains, it is not difficult to see that
Stone algebras form a binary discriminator variety with binary discriminator term x \ y = x ∩ y*.
For a, b ∈ A let a ∨ b be b ∪ (a ∩ b*) and a ∧ b be a ∩ (a ∩ b*)*. Then by theorem above, the
derived algebra (A; ∨, ∧, \, 0) is a left-handed skew Boolean algebra. £
269
