Left reductive regular semigroups

TL;DR

This paper develops a new ideal structure theory for left reductive regular semigroups, establishing a category equivalence with connected categories and applying it to various subclasses and concrete examples.

Contribution

It introduces connected categories as a new framework and proves a category equivalence with left reductive regular semigroups, enabling new structural insights.

Findings

Category equivalence between left reductive regular semigroups and connected categories.

Construction methods for specific subclasses like unipotent and inverse semigroups.

Concrete descriptions for categories from finite transformation and linear transformation semigroups.

Abstract

In this paper we develop an ideal structure theory for the class of left reductive regular semigroups and apply it to several subclasses of popular interest. In these classes we observe that the right ideal structure of the semigroup is `embedded' inside the left ideal one, and so we can construct these semigroups starting with only one object (unlike in other more general cases). To this end, we introduce an upgraded version of Nambooripad's normal category as our building block, which we call a connected category. The main theorem of the paper describes a category equivalence between the category of left (and right) reductive regular semigroups and the category of connected categories. Then, we specialise our result to describe constructions of L- (and R-) unipotent semigroups, right (and left) regular bands, inverse semigroups and arbitrary regular monoids. Finally, we provide concrete (and rather simple) descriptions to the connected categories that arise from finite transformation semigroups, linear transformation semigroups (over a finite dimensional vector space) and symmetric inverse monoids.