Nambooripad's theory of cross-connections

Последнее изменение: 17/12/2019 11:22:44

In the 1930s, J. von Neumann introduced regular rings in his groundbreaking work on von Neumann algebras and continuous geometry. In the early 1970s, the underlying semigroup structure of regular rings naturally became a hot topic in the blossoming area of the algebraic theory of semigroups.

Regular semigroups are very natural objects: they include the semigroup of all functions on an arbitrary set, the multiplicative semigroup Mn(F) of all n x n matrices over a field, indeed the multiplicative semigroup of any regular ring, semilattices, etc. There were several attempts from various directions to give a satisfactory structure theorem for such a rich class of semigroups. But the real obstacle was the complicated idempotent structure of these rather general algebraic objects.

In 1973, Nambooripad, a young Indian mathematician, burst onto this scene with his Ph.D. thesis in which he gave an axiomatic description of the set of idempotents of a (regular) semigroup as a (regular) biordered set. The results were later published in 1979 as A.M.S. Memoirs wherein Nambooripad constructed arbitrary regular semigroups using inductive groupoids (specialized small groupoids whose identities form a regular biordered set) and proved a category equivalence between the category of regular semigroups and the category of inductive groupoids.

An alternative promising approach was initiated by Hall and Grillet around the same time, using the notion of two ‘regular’ partially ordered sets (called regular posets), to build a fundamental regular semigroup. This involved developing the deep notion of cross-connections using which one could describe the explicit relationship between the two regular posets of a regular semigroup. In 1978, Nambooripad proved the equivalence of a pair of cross-connected regular posets and a regular biordered set. Elaborating on that equivalence, Nambooripad extended Grillet’s cross-connection construction to arbitrary regular semigroups by replacing regular posets with normal categories. Normal categories are essentially small categories whose identities form a regular poset. In this way Nambooripad succeeded in proving a category equivalence between the category of regular semigroups and the category of cross-connected normal categories.

Collection of preprints, references and links on Nambooripad's theories

Scanned copy of Nambooripad's treatise on cross-connections Publication No. 28, CMS, Tvm, India, 1994. (Beware of typos!)

V. N. Krishnachandran's notes on the cross-connection theory may be found here (No. 69, preprint)

Scanned copy of Nambooripad's initial version of cross-connections Publication No. 15, CMS, Tvm, India, 1989.
(Beware of more typos and minor errors! But it has an additional section on the applications of the theory.)

Nambooripad's further work on set-based categories Link 1 Link 2

Grillet's papers on cross-connections Paper 1 Paper 2 Paper 3

Nambooripad on relations between Grillet's cross-connections and biordered sets

Nambooripad's seminal work (AMS memoirs) on biordered sets and inductive groupoids

Nambooripad's original Ph. D. thesis where he introduced biordered sets.

Notes on inductive groupoids by E. Krishnan and K. S. S. Nambooripad

Nambooripad's work on locally inverse semigroups (appeared in Simon Stevin, currently unavailable)
I. Pseudo-semilattices and biordered sets II. Pseudo-inverse semigroups III. Regular locally testable semigroups

D. Rajendran's work on bilinear forms Paper 1 Paper 2

P. G. Romeo's Ph. D. thesis on Cross-connections of concordant semigroups

A. R. Rajan's and Sunny Lukose's paper on Ring of normal cones

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