On algebras of P -Ehresmann semigroups and their associate partial semigroups

TLDR: In this article, it was shown that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup, if its projection set is principally finite, then the semigroup algebra of the semi-constant semigroup can be isomorphic to the category algebra of a corresponding category.

Abstract: P-Ehresmann semigroups are introduced by Jones as a common generalization of Ehresmann semigroups and regular $$*$$ -semigroups. Ehresmann semigroups and their semigroup algebras are investigated by many authors in literature. In particular, Stein shows that under some finiteness condition, the semigroup algebra of an Ehresmann semigroup with a left (or right) restriction condition is isomorphic to the category algebra of the corresponding Ehresmann category. In this paper, we generalize this result to P-Ehresmann semigroups. More precisely, we show that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup $$\mathbf{S}$$ , if its projection set is principally finite, then we can give an algebra isomorphism between the semigroup algebra of $$\mathbf{S}$$ and the partial semigroup algebra of the associate partial semigroup of $$\mathbf{S}$$ . Some interpretations and necessary examples are also provided to show why the above isomorphism dose not work for more general P-Ehresmann semigroups.