  $E$-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 .
