Page 139 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 139
IV: Skew Boolean Algebras
that reflect what occurs on 3R, either as a skew lattice or as a skew Boolean algebra, then along
with Theorem 4.3.3R, we are led to the following result of Cvetko-Vah, Leech and Spinks:
Theorem 4.3.4. An identity in [] and ∨ holds in all right-handed skew Boolean algebras,
or more broadly in all right-handed strongly distributive skew lattices, if and only if it holds in
3*. Thus, the question of whether an identity in [] and ∨ holds in these classes of algebras, or in
particular in all partial function algebras of the form P(A, B) is decidable.
Proof. The condition is clearly necessary. Since every right-handed SBA [strongly distributive
skew lattice] is a subalgebra of a power of 3R [under the appropriate signature] the condition is
also sufficient. The final statement is now clear. £
Our interest in such identities and especially in [ ] is due to the role of the latter in right-
handed strongly distributive skew lattices. Returning to Section 2.7, the free categorical and
symmetric right-handed skew lattice on generators x and y can be presented as follows:
x ∨ y −R y ∨ x x[y] = x ∧ (y ∨ x) = (x ∧ y) ∨ x.
y[x] = y ∧ (x ∨ y) = (y ∧ x) ∨ y.
x −R x[y] y[x] −R y where
y ∧ x −R x ∧ y
In particular, this skew diamond and its subalgebras describe the 2-generator subalgebras that can
occur in right-handed skew Boolean algebras or in right-handed strongly distributive skew lattices
in general or, for that matter, in right-handed skew lattices in rings. Thus, just as skew
joins/overrides can be seen as biased unions, and meets as biased intersections, the two other
terms in this diagram besides x and y can be viewed as the outcomes of updating x and y relative
to each other. More will be said about the update operation in Section 5.1.
Axioms for (right-handed) skew Boolean algebras
In their paper, Berendsen et al introduced five identities that describe the behavior of ▹, –
and ∅ in combination. Using automated reasoning software they showed their equations to be
independent and obtained a number of derived equations. They also raised questions about the
algebras that satisfy these equations.
In a responding paper by Cvetko-Vah, Leech and Spinks (Skew lattices and binary
operations on functions, Journal of Applied Logic, 11 (2013), pp. 253-265), the variety of
algebras satisfying those five identities was shown to be term equivalent to the variety of right-
handed skew Boolean algebras. Thus the five identities given in the first paper lead to the
following equational basis (or set or characterizing identities) for right-handed skew Boolean
algebras.
137
that reflect what occurs on 3R, either as a skew lattice or as a skew Boolean algebra, then along
with Theorem 4.3.3R, we are led to the following result of Cvetko-Vah, Leech and Spinks:
Theorem 4.3.4. An identity in [] and ∨ holds in all right-handed skew Boolean algebras,
or more broadly in all right-handed strongly distributive skew lattices, if and only if it holds in
3*. Thus, the question of whether an identity in [] and ∨ holds in these classes of algebras, or in
particular in all partial function algebras of the form P(A, B) is decidable.
Proof. The condition is clearly necessary. Since every right-handed SBA [strongly distributive
skew lattice] is a subalgebra of a power of 3R [under the appropriate signature] the condition is
also sufficient. The final statement is now clear. £
Our interest in such identities and especially in [ ] is due to the role of the latter in right-
handed strongly distributive skew lattices. Returning to Section 2.7, the free categorical and
symmetric right-handed skew lattice on generators x and y can be presented as follows:
x ∨ y −R y ∨ x x[y] = x ∧ (y ∨ x) = (x ∧ y) ∨ x.
y[x] = y ∧ (x ∨ y) = (y ∧ x) ∨ y.
x −R x[y] y[x] −R y where
y ∧ x −R x ∧ y
In particular, this skew diamond and its subalgebras describe the 2-generator subalgebras that can
occur in right-handed skew Boolean algebras or in right-handed strongly distributive skew lattices
in general or, for that matter, in right-handed skew lattices in rings. Thus, just as skew
joins/overrides can be seen as biased unions, and meets as biased intersections, the two other
terms in this diagram besides x and y can be viewed as the outcomes of updating x and y relative
to each other. More will be said about the update operation in Section 5.1.
Axioms for (right-handed) skew Boolean algebras
In their paper, Berendsen et al introduced five identities that describe the behavior of ▹, –
and ∅ in combination. Using automated reasoning software they showed their equations to be
independent and obtained a number of derived equations. They also raised questions about the
algebras that satisfy these equations.
In a responding paper by Cvetko-Vah, Leech and Spinks (Skew lattices and binary
operations on functions, Journal of Applied Logic, 11 (2013), pp. 253-265), the variety of
algebras satisfying those five identities was shown to be term equivalent to the variety of right-
handed skew Boolean algebras. Thus the five identities given in the first paper lead to the
following equational basis (or set or characterizing identities) for right-handed skew Boolean
algebras.
137
