Page 270 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 270
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof of The Bisection Theorem (7.1.1) completed. Given (S; ∨, ∧, \, 0) satisfying (i) – (iii) of
Theorem 7.1.1, consider first the left-handed algebra (S; ∧L, ∨R, \, 0) where x∧Ly = x∧y∧x and
x∨Ry = y∨x∨y. (S; ∨L, ∧L, \, 0) satisfies (i) – (iii) in 7.1.1L making it a left-handed skew Boolean
algebra. Thus (S; ∨, ∧, \, 0), with the identical partial order, is a skew Boolean algebra. £
A stronger version of the above theorem exists; it is Theorem 3.3.21 in Spinks’ 2002
dissertation. Its proof, closely modeled after the one above, is left to the reader.
Theorem 7.1.8. (Spinks [2002]) An algebra (S; ∨, ∧, \, 0) of type <2, 2, 2, 0> is a skew
Boolean algebra if and only if the following conditions hold:
i) (S; ∨, ∧) is a symmetric skew lattice;
ii) (S; \, 0) is an iBCS-algebra;
iii) x ∧ y ∧ x ≈ x \ (x \ y) holds.
In particular, under these conditions, \ is the skew Boolean algebra difference. £
When a binary discriminator variety is also additive
A binary operation + on an algebra A with a constant element 0 is additive if for all a in
A, a + 0 = a = 0 + a. An algebra A with a constant element 0 is additive if an additive operation
can be polynomial defined on A. A variety with a constant 0 is said to be additive if it a binary
term x + y can be polynomial defined satisfying the identities x + 0 ≈ x ≈ 0 + x.
Theorem 7.1.9. If V is a binary discriminator variety with constant term 0 and additive
term x + y, then every algebra A of V has a left-handed skew Boolean algebra term reduct AS.
Proof: By Theorem 7.1.6 the binary term x ∧ y = x \ (x \ y) induces a left normal band operation
on every member of V. Let x + y be the additive term of V and define x ∨ y to be the term y + (x
\ y). We need to show that for any A ∈ V, AS = (A; ∨, ∧, \, 0) is a left-handed skew Boolean
algebra. It is sufficient to show that the left-handed skew Boolean algebra identities hold on any
member A of V on which x \ y induces the binary discriminator. (That is, the primitive case.)
This is done directly by straightforward case-splitting arguments. We consider the associativity
of ∨. Let A be a member of V for which \ is the discriminator. Given a, b, c ∈ A,
a ∨ (b ∨ c) = a ∨ (c + (b \ c)) = (c + (b \ c)) + (a \ (c + (b \ c)))
and similarly
(a ∨ b) ∨ c = c + ((b + (a \ b)) \ c).
Denoting the two expressions on the right above in succession by (L) and (R), we consider four
cases that together cover all possibilities.
268
Proof of The Bisection Theorem (7.1.1) completed. Given (S; ∨, ∧, \, 0) satisfying (i) – (iii) of
Theorem 7.1.1, consider first the left-handed algebra (S; ∧L, ∨R, \, 0) where x∧Ly = x∧y∧x and
x∨Ry = y∨x∨y. (S; ∨L, ∧L, \, 0) satisfies (i) – (iii) in 7.1.1L making it a left-handed skew Boolean
algebra. Thus (S; ∨, ∧, \, 0), with the identical partial order, is a skew Boolean algebra. £
A stronger version of the above theorem exists; it is Theorem 3.3.21 in Spinks’ 2002
dissertation. Its proof, closely modeled after the one above, is left to the reader.
Theorem 7.1.8. (Spinks [2002]) An algebra (S; ∨, ∧, \, 0) of type <2, 2, 2, 0> is a skew
Boolean algebra if and only if the following conditions hold:
i) (S; ∨, ∧) is a symmetric skew lattice;
ii) (S; \, 0) is an iBCS-algebra;
iii) x ∧ y ∧ x ≈ x \ (x \ y) holds.
In particular, under these conditions, \ is the skew Boolean algebra difference. £
When a binary discriminator variety is also additive
A binary operation + on an algebra A with a constant element 0 is additive if for all a in
A, a + 0 = a = 0 + a. An algebra A with a constant element 0 is additive if an additive operation
can be polynomial defined on A. A variety with a constant 0 is said to be additive if it a binary
term x + y can be polynomial defined satisfying the identities x + 0 ≈ x ≈ 0 + x.
Theorem 7.1.9. If V is a binary discriminator variety with constant term 0 and additive
term x + y, then every algebra A of V has a left-handed skew Boolean algebra term reduct AS.
Proof: By Theorem 7.1.6 the binary term x ∧ y = x \ (x \ y) induces a left normal band operation
on every member of V. Let x + y be the additive term of V and define x ∨ y to be the term y + (x
\ y). We need to show that for any A ∈ V, AS = (A; ∨, ∧, \, 0) is a left-handed skew Boolean
algebra. It is sufficient to show that the left-handed skew Boolean algebra identities hold on any
member A of V on which x \ y induces the binary discriminator. (That is, the primitive case.)
This is done directly by straightforward case-splitting arguments. We consider the associativity
of ∨. Let A be a member of V for which \ is the discriminator. Given a, b, c ∈ A,
a ∨ (b ∨ c) = a ∨ (c + (b \ c)) = (c + (b \ c)) + (a \ (c + (b \ c)))
and similarly
(a ∨ b) ∨ c = c + ((b + (a \ b)) \ c).
Denoting the two expressions on the right above in succession by (L) and (R), we consider four
cases that together cover all possibilities.
268
