The idea of using small categories to give structural descriptions of semigroups goes back to the famous ESN (Ehresmann-Schein-Nambooripad) theorem. Among other things, this leads to the ideal structure description of regular semigroups using a cross-connected pair of categories by Nambooripad. But in the class of left reductive semigroups (which includes many ‘popular’ classes of semigroups like monoids, inverse semigroups etc.), we observe that the right ideal structure of the semigroups is ‘embedded’ inside the left ideal one. So we can construct these semigroups starting with only a single category, unlike in the more general cases.
In this talk, we give an overview of Nambooripad’s cross-connection of normal categories and further develop the ideal structure theory for the class of left reductive regular semigroups using a new modified approach. To this end, we introduce an upgraded version of Nambooripad’s normal category as our building block, which we call a connected category. This is a recent joint work with Gracinda Gomes.