Jonathan Leech

DUAL SYMMETRIC INVERSE MONOIDS AND REPRESENTATION THEORY

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid I_X , that is, a monoid of partial one-to-one self-maps of a set X. The present paper describes the structure of a categorical dual I*_X to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in I_X and I*_X, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

Skew Boolean algebras and discriminator varieties

We investigate the class of skew Boolean algebras which are also meet semilattices under the natural skew lattice partial order. Such algebras, called hereskew Boolean ∩-algebras, are quite common. Indeed, any algebra A in a discriminator variety with a constant term has a skew Boolean ∩-algebra polynomial reduct whose congruences coincide with those of A.

The geometric structure of skew lattices

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 ob- ject. 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. Var- ious 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 countermultipli- cation. 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 con- nected 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 ana- logue 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 (10-12) 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 deter- mined 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 refer- ence 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-

Cancellation in Skew Lattices

Distributive lattices are well known to be precisely those lattices that possess cancellation:

Categorical Skew Lattices

x \lor y = x \lor z

Skew lattices and binary operations on functions

and

Rings Whose Idempotents Form a Multiplicative Set

x \land y = x \land z

Extending Groups by Monoids

imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M3 or N5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations

ON MAXIMAL IDEMPOTENT-CLOSED SUBRINGS OF Mn(𝔽)

\land

Free skew Boolean algebras

and