Algebras of Ehresmann semigroups and categories

$E$-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an $E$-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize $E$-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples. Erratum: Shoufeng Wang discovered an error in the main theorem of the paper. Wang observed that the function we suggest as an isomorphism is not a homomorphism unless the semigroup being discussed is left restriction. In order to fix our mistake we will add this assumption. Note that our revised result is still a generalization of earlier work of Guo and Chen, the author, and Steinberg .