On algebras of <Emphasis Type="Italic">P</Emphasis>-Ehresmann semigroups and their associate partial semigroups

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.