Page 123 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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IV: Skew Boolean Algebras
In the seventh and final chapter, we examine further the role of skew Boolean algebras in
universal algebra, in particular in the study of what might be termed “generalized Boolean
phenomena,” a topic of continuing interest in universal algebra. In particular we will present a
number of results by Robert Bignall and his student, Matthew Spinks. Both gentlemen have
made significant contributions in this area.
This chapter concludes with a discussion of historical aspects of this topic and references.
4.1 Skew Boolean algebras
We have seen that any maximal normal band S in a ring R is a skew lattice in R under
multiplication and the cubic join ∇. Indeed, every such band is the full set of idempotents in the
subring it generates in R. There is more.
To begin, if E(R) is commutative, then E(R) is a generalized Boolean lattice since for
each e in E(R), the principal lattice ideal ⎡e⎤ = {f⎮f ≤ e} is a Boolean lattice. Indeed, if e ≥ f in
E(R), then e – f ∈ E(R) also; moreover e ≥ e – f with f ∧ (e – f) = 0 and f ∨ ( e – f) = e. All this is
true even when E(R) is not commutative, provided it forms a band under multiplication. Since
E(R) is then a normal skew lattice by Theorem 2.3.7, each ⎡e⎤ is a sublattice; what is more, for all
f ∈ ⎡e⎤ one has e – f ∈ ⎡e⎤ with f ∨ (e – f) = e and f ∧ (e – f) = 0. This leads us to a definition:
A skew Boolean algebra is an algebra (S; ∨, ∧, \, 0) such that
(i) (S; ∨, ∧, 0) is a strongly distributive skew lattice with 0 and
(ii) \ is a binary operation satisfying (e∧f∧e) ∨ (e\f) = e and (e∧f∧e) ∧ (e\f) = 0.
By Theorem 2.3.4, (i) is equivalent to (S; ∨, ∧) being distributive, symmetric and normal and
having the 0-identities hold. (i) and (ii) together imply e∧f∧e and e\f commute. They also imply
that each ⎡e⎤ is a Boolean sublattice of S with e\f being the unique complement of e∧f∧e in ⎡e⎤.
Conversely given (i), if each [e] forms a Boolean lattice, then (ii) follows. These observations
give:
Theorem 4.1.1. Skew Boolean algebras form a variety; moreover, every congruence on
the skew lattice reduct of a skew Boolean algebra is also a skew Boolean algebra congruence. In
particular, the Green’s equivalences are all skew Boolean algebra congruences. £
This variety will be denoted by SBA. The broader perspective of skew lattices yields:
Theorem 4.1.2. A normal, symmetric skew lattice with 0 forms a skew Boolean algebra
if and only if its maximal lattice image S/D is a generalized Boolean lattice, in which case \ is
implicitly determined by (ii) above.
121
In the seventh and final chapter, we examine further the role of skew Boolean algebras in
universal algebra, in particular in the study of what might be termed “generalized Boolean
phenomena,” a topic of continuing interest in universal algebra. In particular we will present a
number of results by Robert Bignall and his student, Matthew Spinks. Both gentlemen have
made significant contributions in this area.
This chapter concludes with a discussion of historical aspects of this topic and references.
4.1 Skew Boolean algebras
We have seen that any maximal normal band S in a ring R is a skew lattice in R under
multiplication and the cubic join ∇. Indeed, every such band is the full set of idempotents in the
subring it generates in R. There is more.
To begin, if E(R) is commutative, then E(R) is a generalized Boolean lattice since for
each e in E(R), the principal lattice ideal ⎡e⎤ = {f⎮f ≤ e} is a Boolean lattice. Indeed, if e ≥ f in
E(R), then e – f ∈ E(R) also; moreover e ≥ e – f with f ∧ (e – f) = 0 and f ∨ ( e – f) = e. All this is
true even when E(R) is not commutative, provided it forms a band under multiplication. Since
E(R) is then a normal skew lattice by Theorem 2.3.7, each ⎡e⎤ is a sublattice; what is more, for all
f ∈ ⎡e⎤ one has e – f ∈ ⎡e⎤ with f ∨ (e – f) = e and f ∧ (e – f) = 0. This leads us to a definition:
A skew Boolean algebra is an algebra (S; ∨, ∧, \, 0) such that
(i) (S; ∨, ∧, 0) is a strongly distributive skew lattice with 0 and
(ii) \ is a binary operation satisfying (e∧f∧e) ∨ (e\f) = e and (e∧f∧e) ∧ (e\f) = 0.
By Theorem 2.3.4, (i) is equivalent to (S; ∨, ∧) being distributive, symmetric and normal and
having the 0-identities hold. (i) and (ii) together imply e∧f∧e and e\f commute. They also imply
that each ⎡e⎤ is a Boolean sublattice of S with e\f being the unique complement of e∧f∧e in ⎡e⎤.
Conversely given (i), if each [e] forms a Boolean lattice, then (ii) follows. These observations
give:
Theorem 4.1.1. Skew Boolean algebras form a variety; moreover, every congruence on
the skew lattice reduct of a skew Boolean algebra is also a skew Boolean algebra congruence. In
particular, the Green’s equivalences are all skew Boolean algebra congruences. £
This variety will be denoted by SBA. The broader perspective of skew lattices yields:
Theorem 4.1.2. A normal, symmetric skew lattice with 0 forms a skew Boolean algebra
if and only if its maximal lattice image S/D is a generalized Boolean lattice, in which case \ is
implicitly determined by (ii) above.
121
