Chain projection ordered categories and 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].