Page 14 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 14
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
A second source of motivation arises from studying the multiplicative semigroups of
rings. In the study of rings, idempotents play an important role. In general, given idempotents e
and f in a ring, their product ef need not be idempotent (unless, e.g., the ring is commutative).
Nonetheless ef is “below and to the right” of e in that e(ef) = ef, and at the same time “below and
to the left” of f in that (ef)f = ef. Dually, e is “below and to the left” of the circle product e¡f = e
+ f – ef in that e(e¡f) = e, while f is “below and to the right” of e¡f in that (e¡f)f = f. On thus has
the following picture:
e¡f
ef
ef.
In general, given an element x in a ring, x2 = x iff x¡x = x. What is more, for a set of
idempotents in a ring that is closed under both multiplication and ¡, the following four absorption
identities are satisfied.
a(a ¡ b) = a = (b ¡ a)a. a ¡ (ab) = a = (ba) ¡ a.
Noncommutative rings that are well endowed with idempotents are rich in such examples.
What can one say about their structure? Such ring-based structures will occupy our attention in
much of the second and sixth chapters to follow.
Abstracting only slightly, one can think of bands – semigroups of idempotents – that are
rich enough in structure to possess an idempotent counter-multiplication. Thus multiplication
produces products that are generally “further down” in the band, while the counter-products
would be generally “further up” in the band. Such bands exist. What can be said about them?
A third source of motivation comes from universal algebra, especially the study of what
may be loosely termed “generalized Boolean phenomena”. Do noncommutative generalizations
of (generalized) Boolean lattices and algebras exist? If so, what connections exist between them
and other structures related to Boolean algebras? Clearly noncommutative lattice theory have
something to say about all this? Questions such as these will occupy our attention in the fourth
and seventh chapters to follow.
In the meanwhile, we begin this introductory chapter by reviewing a number of concepts
about lattices and universal algebra in the first section. In the second section we recall various
facts about bands that are pertinent to the rest of the monograph. And then in Section 3 we discuss
some “first principals” of noncommutative lattices. All the material in this chapter is foundational
to what follows later. The reader well-versed in the material in either of the first two sections can
easily skip over one or both of them and then proceed to the third section. We emphasize,
however, that a firm grasp of regular bands, and their left and right-sided cases, is crucial to
understanding much that will be said about skew lattices.
Left regular bands have received increased attention recently due to their role in
combinatorial aspects of algebra and geometry. See, e.g., the introductory remarks in the
12
A second source of motivation arises from studying the multiplicative semigroups of
rings. In the study of rings, idempotents play an important role. In general, given idempotents e
and f in a ring, their product ef need not be idempotent (unless, e.g., the ring is commutative).
Nonetheless ef is “below and to the right” of e in that e(ef) = ef, and at the same time “below and
to the left” of f in that (ef)f = ef. Dually, e is “below and to the left” of the circle product e¡f = e
+ f – ef in that e(e¡f) = e, while f is “below and to the right” of e¡f in that (e¡f)f = f. On thus has
the following picture:
e¡f
ef
ef.
In general, given an element x in a ring, x2 = x iff x¡x = x. What is more, for a set of
idempotents in a ring that is closed under both multiplication and ¡, the following four absorption
identities are satisfied.
a(a ¡ b) = a = (b ¡ a)a. a ¡ (ab) = a = (ba) ¡ a.
Noncommutative rings that are well endowed with idempotents are rich in such examples.
What can one say about their structure? Such ring-based structures will occupy our attention in
much of the second and sixth chapters to follow.
Abstracting only slightly, one can think of bands – semigroups of idempotents – that are
rich enough in structure to possess an idempotent counter-multiplication. Thus multiplication
produces products that are generally “further down” in the band, while the counter-products
would be generally “further up” in the band. Such bands exist. What can be said about them?
A third source of motivation comes from universal algebra, especially the study of what
may be loosely termed “generalized Boolean phenomena”. Do noncommutative generalizations
of (generalized) Boolean lattices and algebras exist? If so, what connections exist between them
and other structures related to Boolean algebras? Clearly noncommutative lattice theory have
something to say about all this? Questions such as these will occupy our attention in the fourth
and seventh chapters to follow.
In the meanwhile, we begin this introductory chapter by reviewing a number of concepts
about lattices and universal algebra in the first section. In the second section we recall various
facts about bands that are pertinent to the rest of the monograph. And then in Section 3 we discuss
some “first principals” of noncommutative lattices. All the material in this chapter is foundational
to what follows later. The reader well-versed in the material in either of the first two sections can
easily skip over one or both of them and then proceed to the third section. We emphasize,
however, that a firm grasp of regular bands, and their left and right-sided cases, is crucial to
understanding much that will be said about skew lattices.
Left regular bands have received increased attention recently due to their role in
combinatorial aspects of algebra and geometry. See, e.g., the introductory remarks in the
12
