Connections beyond cross-connections

Abstract : The talk will be on the structure theory of regular semigroups with special emphasis on the theory of cross-connections. A semigroup S is said to be (von Neumann) regular if for every a 2 S, there exists b such that aba = a. Cross-connection is an abstract construction of regular semigroups from ‘normal categories ’. Although the formal study of semigroups began in the early 20th century, structure theorems in semigroups were very scarce due to the obvious limitations imposed by the generality of the structure. So the focus shifted to describing structures for special classes like inverse semigroups, regular semigroups etc and one of the major breakthrough in this context was the characterization of the idempotents of a semigroup as a bi-ordered set by K.S.S. Nambooripad in the early 1970s. Later there were attempts to describe the structure of semigroups from its ideals and in this context, P. A. Grillet introduced the notion of cross-connections and constructed the fundamental image of a regular semigroup as a cross-connection semigroup from the regular partially ordered sets of principal ideals. In 1990s, Nambooripad extended this to arbitrary regular semigroups by characterizing the principal ideals of a semigroup as normal categories. In the talk, we will briefly discuss the basic concepts and developments leading to the cross-connection theory. And then we will consider an explicit construction of a cross-connection semigroup starting from an arbitrary group G and two sets I and ?. Using this, we describe the structure of completely simple semigroups and this shows that the cross-connection here plays the role of the so-called structure matrix of a completely simple semigroup. We also discuss the notion of duality in the theory of cross-connections and using the semigroup of endomorphisms of a vectorspace V, we show how this duality coincides with the conventional notion of duality of vectorspaces. We see that this observation and certain other encouraging works related to Fredholm operators will lead us naturally to the domain of operator algebras and regular rings. We conclude by discussing further scope of the cross-connection theory and related techniques of algebraic theory of semigroups in these contexts and how it may be connected to other branches of mathematics like C-* algebras, von Neumann algebras, regular rings, manifolds etc.