Page 162 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 162
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
project with Leech published in 2007. Further developments of that material occur in a 2013
paper by Kudryatseva, who is also publishing a paper on free skew Boolean intersection algebras,
set to appear in 2017.
Bignall, Cornish and Leech were not the only ones to initiate a study of noncommutative
Boolean algebras in some form. As seen in Section 4.2, computer scientists (J. Berendsen et al)
in studying the override and update operations introduced in 2010 a class of algebras that was
shown by Cvetko-Vah, Leech and Spinks in their 2013 paper to be term equivalent to right-
handed skew Boolean algebras. Indeed they are (∨, \, 0)-term reducts of the latter with a variant
of ∧ obtained as a defined operation. In a 2011 paper, Janis Cirulis, studied near lattices (meet
semilattices with joins existing for pairs of elements with common upper bounds) that were
supplied with an override operation (reducing to the conditional commuting join when it existed).
Under special assumptions Cirulis obtained a class of skew Boolean algebras with intersections,
with ∩ being the near lattice meet.
Skew Boolean algebras clearly form a natural class of objects to study. They play a
significant role in the more general study of Boolean-like phenomena, with connections to
discriminator varieties, iBCK algebras and their offspring which we meet in Chapter 7, and more
recently, Church algebras. (See Spinks [2002] and Cvetko-Vah and Salibra [2015]) It is no
coincidence that skew Boolean algebras have attracted the interest of some in computer science.
Indeed, most of the individuals mentioned in these paragraphs have some degree of research
interest in computer science. (In particular, see Bignall and Spinks [1996] – [1998] and Spinks
and Veroff. [2006])
Topological representations of (generalized) Boolean algebras and distributive lattices
have been studied since M. H. Stone’s papers in the 1930s. More recently this has been extended
to skew Boolean algebras (possibly with intersections) and strongly distributive skew lattices.
Here one is given a skew Boolean algebra S with maximal (generalized) Boolean image B = S/D.
The latter is dual to a topological space B under standard Stone duality with S itself being dual to
an étale covering π: X → B. (See the papers below by Bauer and Cvetko-Vah (et al) and by
Kudryatseva.)
In Chapter 6 we will look at the skew Boolean algebras of idempotents in rings, and in
particular, the case where the idempotents in a ring are closed under multiplication and thus
naturally form a skew Boolean algebra. In Chapter 7 we will look at algebraic structures that
support a skew lattice reduct, in some cases with intersections; that is, we will look at algebras
where skew Boolean operations are term-defined using the given operations of the algebra.
160
project with Leech published in 2007. Further developments of that material occur in a 2013
paper by Kudryatseva, who is also publishing a paper on free skew Boolean intersection algebras,
set to appear in 2017.
Bignall, Cornish and Leech were not the only ones to initiate a study of noncommutative
Boolean algebras in some form. As seen in Section 4.2, computer scientists (J. Berendsen et al)
in studying the override and update operations introduced in 2010 a class of algebras that was
shown by Cvetko-Vah, Leech and Spinks in their 2013 paper to be term equivalent to right-
handed skew Boolean algebras. Indeed they are (∨, \, 0)-term reducts of the latter with a variant
of ∧ obtained as a defined operation. In a 2011 paper, Janis Cirulis, studied near lattices (meet
semilattices with joins existing for pairs of elements with common upper bounds) that were
supplied with an override operation (reducing to the conditional commuting join when it existed).
Under special assumptions Cirulis obtained a class of skew Boolean algebras with intersections,
with ∩ being the near lattice meet.
Skew Boolean algebras clearly form a natural class of objects to study. They play a
significant role in the more general study of Boolean-like phenomena, with connections to
discriminator varieties, iBCK algebras and their offspring which we meet in Chapter 7, and more
recently, Church algebras. (See Spinks [2002] and Cvetko-Vah and Salibra [2015]) It is no
coincidence that skew Boolean algebras have attracted the interest of some in computer science.
Indeed, most of the individuals mentioned in these paragraphs have some degree of research
interest in computer science. (In particular, see Bignall and Spinks [1996] – [1998] and Spinks
and Veroff. [2006])
Topological representations of (generalized) Boolean algebras and distributive lattices
have been studied since M. H. Stone’s papers in the 1930s. More recently this has been extended
to skew Boolean algebras (possibly with intersections) and strongly distributive skew lattices.
Here one is given a skew Boolean algebra S with maximal (generalized) Boolean image B = S/D.
The latter is dual to a topological space B under standard Stone duality with S itself being dual to
an étale covering π: X → B. (See the papers below by Bauer and Cvetko-Vah (et al) and by
Kudryatseva.)
In Chapter 6 we will look at the skew Boolean algebras of idempotents in rings, and in
particular, the case where the idempotents in a ring are closed under multiplication and thus
naturally form a skew Boolean algebra. In Chapter 7 we will look at algebraic structures that
support a skew lattice reduct, in some cases with intersections; that is, we will look at algebras
where skew Boolean operations are term-defined using the given operations of the algebra.
160
