The Geometric Structure of Skew Lattices

Abstract

Abstract. A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric object. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Various aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators.Recall that a band is a semigroup satisfying the idempotent law: xx = x. Upon examining bands which are multiplicative subsemigroups of rings, one uncovers classes of bands which also possess an idempotent countermultiplication. This leads one to define a skew lattice to be an algebra with a pair of associative idempotent binary operations, the join and the meet, which are connected by a set of absorption laws (see 1.1). While skew lattices of idempotents in rings remain important sources of motivation, the results of [10-12] make it clear that skew lattices can sustain mathematical life on their own. Perhaps the most natural way to think about a skew lattice is as a noncommutative analogue of a lattice. As such, a skew lattice is not only an algebraic object, but also a geometric object. Thus far most of the research given in [10][11][12] has emphasized the algebraic side of skew lattices. The purpose of this paper is to investigate their geometric aspects and in particular the role of the natural partial order in determining their algebraic structure, much as a lattice is determined by its natural partial ordering. It is not our goal, however, to reinvent lattice theory. Hence the geometry of a skew lattice will be studied relative to the fixed structure of its underlying lattice. Saying this entails an implicit reference to the fundamental Clifford-McLean Theorem which in effect provides a first sketch of a skew lattice: a congruence is defined on each skew lattice (called natural equivalence) which induces its maximal lattice image and whose equiv-

Flat coset decompositions of skew lattices

“…This approach enables us to classify certain varieties of skew lattices. The results of Section 4 were motivated by the earlier studies of [17], [5], [21], [7], [22] and [23]. We demonstrate the impact of the study of the flat coset structure in the case of skew lattices of matrices, following the work of [8], [9] and [4].…”

“…In [5] examples are given that show the independence between (i) and (ii) above. Furthermore, the skew lattices satisfying (i) correspond to the lower symmetric skew lattices, while the skew lattices satisfying (ii) correspond to the upper symmetric skew lattices as discussed in the proof of Theorem 3.5 in the paper [17], and expressed below: …”

“…Already in the 1989 foundation paper [15] certain aspects of the coset structure of a skew lattice were analyzed, however it was fully explored in [17] where Leech studied what he referred to as the global geometry of skew lattices. The coset structure was later used in [6] and [7] to characterize certain sub-varieties of skew lattices, and in [14] for the purpose of studying the distributivity of skew lattices, an approach proposed in [23].…”

“…Given a pair of comparable D-classes each of the two classes induces a partition of the other class, and the blocks of these partitions are called cosets. The internal structure of a skew lattice is described by the coset decompositions introduced in [17] that reveal the interplay of a pair of comparable D-classes.…”

“…If T meets each D-class of S in exactly one point, then T is called a lattice section of S. As such, it is a maximal sub-lattice that is also an internal copy inside S of the maximal lattice image S/D [17]. A lattice section of S (if it exists) is therefore isomorphic to S/D.…”