Page 140 - 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
Theorem 4.3.5. An independent set of identities that characterize the variety of all right-
handed skew Boolean algebras (S; ∧, ∨, \, 0) is given by:
i) x ∨ x ≈ x.
ii) x \ x ≈ 0.
iii) x ∨ y ≈ (y \ x) ∨ x.
iv) x \ (y \ z) ≈ (x \ y) ∨ (x \ (x \ z)).
v) (x ∨ y) \ z ≈ (x \ z) ∨ (y \ z).
Here the meet ∧ is defined by x∧y : = y \ (y \ x). £
An independent set of six identities characterizing the variety of skew Boolean algebras
was given in Spinks’ dissertation: (See also Spinks [1998] Prop. LBSL-RC-7.)
i) x∧(x∨y) ≈ x; ii) (y∧x)∨x ≈ x;
iii) x ∧ (y∨z) ≈ (x∧y) ∨ (x∧z); iv) (x∨y) ∧ z ≈ (x∧z) ∨ (y∧z);
v) ((x∧y)∧x) ∨ (x\y) ≈ x; vi) (x\y) ∧ y ≈ 0.
These contrast with the 12 identities implicit in the definition of a skew Boolean algebra in
Section 4.1. As of this writing, neither of these two sets of identities have been bettered with
respect to size. Of interest in the first set is the fact that along with 0 which is just used once, the
binary operations \ and ∨ suffice in their statements.
Could ∧ be eliminated from the second set of 2-sided identities? Actually yes, thanks to
the identity:
x∧y = (y∨x) \ {[(y∨x) \ x] ∨ [(y∨x) \ y]}.
This identity is easily seen to hold on primitive algebras, and hence on all skew Boolean algebras.
Returning to all six identities above, it can be used to first define ∧ in terms of ∨ and \, and then
to eliminate all occurrences of ∧ in the six identities. A reduced signature (∨, \, 0) is thus possible
for skew Boolean algebras, but not convenient. As with generalized Boolean algebras, ∨ cannot
be eliminated in favor of ∧, \ and 0. A closer look at the role of \ occurs in the final chapter.
4.4 Skew Boolean algebras with intersections
Recall that a skew lattice (S; ∨, ∧) has finite intersections if every pair e, f ∈S
possesses a natural meet with respect to the natural partial order ≥ on S. When it occurs,
the natural meet of any e and f in S is denoted by e∩f. We say that (S; ∨, ∧) has
intersections if the natural partial order ≥ on S has infima for arbitrary subsets. In general
the following must hold:
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Theorem 4.3.5. An independent set of identities that characterize the variety of all right-
handed skew Boolean algebras (S; ∧, ∨, \, 0) is given by:
i) x ∨ x ≈ x.
ii) x \ x ≈ 0.
iii) x ∨ y ≈ (y \ x) ∨ x.
iv) x \ (y \ z) ≈ (x \ y) ∨ (x \ (x \ z)).
v) (x ∨ y) \ z ≈ (x \ z) ∨ (y \ z).
Here the meet ∧ is defined by x∧y : = y \ (y \ x). £
An independent set of six identities characterizing the variety of skew Boolean algebras
was given in Spinks’ dissertation: (See also Spinks [1998] Prop. LBSL-RC-7.)
i) x∧(x∨y) ≈ x; ii) (y∧x)∨x ≈ x;
iii) x ∧ (y∨z) ≈ (x∧y) ∨ (x∧z); iv) (x∨y) ∧ z ≈ (x∧z) ∨ (y∧z);
v) ((x∧y)∧x) ∨ (x\y) ≈ x; vi) (x\y) ∧ y ≈ 0.
These contrast with the 12 identities implicit in the definition of a skew Boolean algebra in
Section 4.1. As of this writing, neither of these two sets of identities have been bettered with
respect to size. Of interest in the first set is the fact that along with 0 which is just used once, the
binary operations \ and ∨ suffice in their statements.
Could ∧ be eliminated from the second set of 2-sided identities? Actually yes, thanks to
the identity:
x∧y = (y∨x) \ {[(y∨x) \ x] ∨ [(y∨x) \ y]}.
This identity is easily seen to hold on primitive algebras, and hence on all skew Boolean algebras.
Returning to all six identities above, it can be used to first define ∧ in terms of ∨ and \, and then
to eliminate all occurrences of ∧ in the six identities. A reduced signature (∨, \, 0) is thus possible
for skew Boolean algebras, but not convenient. As with generalized Boolean algebras, ∨ cannot
be eliminated in favor of ∧, \ and 0. A closer look at the role of \ occurs in the final chapter.
4.4 Skew Boolean algebras with intersections
Recall that a skew lattice (S; ∨, ∧) has finite intersections if every pair e, f ∈S
possesses a natural meet with respect to the natural partial order ≥ on S. When it occurs,
the natural meet of any e and f in S is denoted by e∩f. We say that (S; ∨, ∧) has
intersections if the natural partial order ≥ on S has infima for arbitrary subsets. In general
the following must hold:
138
