Left reductive regular semigroups

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. We dub these new categories as connected categories. The main theorem of the paper describes a category equivalence between the category of left reductive regular semigroups and the category of connected categories. Then, we specialise our result to describe constructions of L-unipotent semigroups, right regular bands, inverse semigroups and arbitrary regular monoids. Intriguingly, the same results hold for the dual classes of right reductive regular semigroups, R-unipotent semigroups and left regular bands; this is due to the category isomorphism between the left and right reductive regular semigroups. Finally, we provide concrete and relatively simple descriptions of the connected categories that arise from finite transformation semigroups, linear transformation semigroups (over a finite dimensional vector space) and symmetric inverse monoids.