Chain projection ordered categories and DRC-restriction semigroups

TL;DR

This paper develops a theory of chain projection ordered categories, generalizes existing structures, and establishes a structure theorem linking DRC-restriction semigroups with these categories, providing new insights and applications.

Contribution

It introduces a generalized framework for chain projection ordered categories and proves a structure theorem connecting them to DRC-restriction semigroups, solving a problem posed by East and Azeef Muhammed.

Findings

Category of DRC-restriction semigroups is isomorphic to chain projection ordered categories.

Existence of free projection-generated DRC-restriction semigroups for any strong two-sided projection algebra.

Reestablishment of the structure of projection-fundamental DRC-restriction semigroups.

Abstract

In this paper we provide a theory of chain projection ordered categories and generalize that of chain projection ordered groupoids developed by East and Azeef Muhammed recently. By using chain projection ordered categories, we obtain a structure theorem for DRC-restriction semigroups. More specifically, we prove that the category of DRC-restriction semigroups together with (2,1,1)-homomorphisms is isomorphic to the category of chain projection ordered categories together with chain projection ordered functors. Moreover, some special cases are also considered. As applications of the main theorem, we demonstrate that the existence of free projection-generated DRC-restriction semigroups associated to any strong two-sided projection algebra and reobtain the structures of projection-fundamental DRC-restriction semigroups. Our work may be regarded as an answer for the fourth problem proposed by East and Azeef Muhammed in [Advances in Mathematics, 437 (2024) 109447].