Page 264 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 264
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
7.1 Differences, discriminators and connections with other algebras
Besides a strongly distributive skew lattice reduct (S; ∨, ∧), a skew Boolean algebra also
has a complementary reduct, (S; \, 0). To understand the behavior of the latter, consider the
following identities:
(a) x \ x ≈ 0.
(b) x \ (y \ x) ≈ x.
(c) (x \ y) \ z ≈ (x \ z) \ y.
(d) (x \ y) \ z ≈ (x \ z) \ (y \ z).
(e) x \ 0 ≈ x
(f) 0 \ x ≈ 0,
(g) x \ (x \ y) ≈ y \ (y \ x).
An algebra (S; \, 0) of type (2, 0) satisfying (a) – (d) is called an implicative BCS-algebra (iBCS-
algebra for short), in which case it also satisfies both (e) and (f) making it a 0-subtractive algebra.
Indeed (e) follows by putting y = x in (b) and then using (a), while (f) follows by setting x = 0 in
(b) and then using (e). If in addition (S; \, 0) satisfies the "commutative" identity (g), it is called
an implicative BCK-algebra (iBCK-algebra for short).
iBCK-algebras were introduced in Lyndon [1951]. They have been studied by various
authors including Abbott [1967], Cornish [1982], Iseki and Tanaka [1978] and Kalman [1960].
iBCS-algebras were introduced and studied by Bignall and Spinks in [2003] and [2007]. For
skew Boolean algebras we have the following Signature Bisection Theorem.
Theorem 7.1.1. An algebra (S; ∨, ∧, \, 0) of type 〈2, 2, 2, 0〉 forms a skew Boolean
algebra if and only if:
i) (S; ∨, ∧) is a strongly distributive skew lattice.
ii) (S; \, 0) is an iBCS-algebra.
iii) The identity e \ (e \ f) ≈ e∧f∧e holds.
(S; ∨, ∧, \, 0) is a generalized Boolean algebra if and only if (ii) can be strengthened to:
iiʹ) (S; \, 0) is an iBCK-algebra.
Proof Given skew Boolean algebra, (S; ∨, ∧, \, 0), (i) holds and it is easily seen that the reduct
(S; \, 0) is an iBCS algebra and that e \ (e \ f) ≈ e∧f∧e. Just check the situation for 0-primitive
algebras. Thus (i) - (iii) follow. The converse will be proved after Theorem 1.11 below. Given
(i)-(iii), (S; \, 0) is an iBCK-algebra if and only if identity (g) holds, which in this context is
equivalent to e∧f∧e ≈ f∧e∧f holding. But the latter is equivalent to e∧f ≈ f∧e, making (S; ∨, ∧) a
lattice and (S; ∨, ∧, \, 0) a generalized Boolean algebra. £
To distinguish an iBCK operation from the more general iBCS operation, we use the
symbol / when referring to the former. Given an iBCK algebra (S; /, 0), set x∩y = x/(x/y). Then:
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7.1 Differences, discriminators and connections with other algebras
Besides a strongly distributive skew lattice reduct (S; ∨, ∧), a skew Boolean algebra also
has a complementary reduct, (S; \, 0). To understand the behavior of the latter, consider the
following identities:
(a) x \ x ≈ 0.
(b) x \ (y \ x) ≈ x.
(c) (x \ y) \ z ≈ (x \ z) \ y.
(d) (x \ y) \ z ≈ (x \ z) \ (y \ z).
(e) x \ 0 ≈ x
(f) 0 \ x ≈ 0,
(g) x \ (x \ y) ≈ y \ (y \ x).
An algebra (S; \, 0) of type (2, 0) satisfying (a) – (d) is called an implicative BCS-algebra (iBCS-
algebra for short), in which case it also satisfies both (e) and (f) making it a 0-subtractive algebra.
Indeed (e) follows by putting y = x in (b) and then using (a), while (f) follows by setting x = 0 in
(b) and then using (e). If in addition (S; \, 0) satisfies the "commutative" identity (g), it is called
an implicative BCK-algebra (iBCK-algebra for short).
iBCK-algebras were introduced in Lyndon [1951]. They have been studied by various
authors including Abbott [1967], Cornish [1982], Iseki and Tanaka [1978] and Kalman [1960].
iBCS-algebras were introduced and studied by Bignall and Spinks in [2003] and [2007]. For
skew Boolean algebras we have the following Signature Bisection Theorem.
Theorem 7.1.1. An algebra (S; ∨, ∧, \, 0) of type 〈2, 2, 2, 0〉 forms a skew Boolean
algebra if and only if:
i) (S; ∨, ∧) is a strongly distributive skew lattice.
ii) (S; \, 0) is an iBCS-algebra.
iii) The identity e \ (e \ f) ≈ e∧f∧e holds.
(S; ∨, ∧, \, 0) is a generalized Boolean algebra if and only if (ii) can be strengthened to:
iiʹ) (S; \, 0) is an iBCK-algebra.
Proof Given skew Boolean algebra, (S; ∨, ∧, \, 0), (i) holds and it is easily seen that the reduct
(S; \, 0) is an iBCS algebra and that e \ (e \ f) ≈ e∧f∧e. Just check the situation for 0-primitive
algebras. Thus (i) - (iii) follow. The converse will be proved after Theorem 1.11 below. Given
(i)-(iii), (S; \, 0) is an iBCK-algebra if and only if identity (g) holds, which in this context is
equivalent to e∧f∧e ≈ f∧e∧f holding. But the latter is equivalent to e∧f ≈ f∧e, making (S; ∨, ∧) a
lattice and (S; ∨, ∧, \, 0) a generalized Boolean algebra. £
To distinguish an iBCK operation from the more general iBCS operation, we use the
symbol / when referring to the former. Given an iBCK algebra (S; /, 0), set x∩y = x/(x/y). Then:
262
