Algebras of Ehresmann semigroups and categories

Abstract

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 EEhresmann 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 to: Algebras of Ehresmann semigroups and categories

“…Proof of Theorem 1.5 The proof that ϕ and ψ are bijections is identical to that in [3], as is Case 1 of the proof that ϕ is a homomorphism.…”

“…Take x ≤ x such that x x and assume that there is a y ≤ y such that ∃x · y , that is, r(x ) = d(y ). Since y ≤ y we have r(x ) = d(y ) ≤ d(y) by [3,CO1]. Now, by [3,EC7] | r(x) ∧ d(y)) =x and r(x ) ∧ d(y) = r(x ) so we get …”

“…This Erratum shows how this assumption yields a valid result, and examines the consequences. We assume the reader is familiar with [3] and in particular with the definition of an E-Ehresmann semigroup; for undefined terms, the reader should consult [3].…”

“…We start by giving a counterexample to the original claim in [3,Theorem 3.4]. Let S be an E-Ehresmann semigroup with ≤=≤ r a principally finite poset, and let C = C(S) be the corresponding Ehresmann category.…”

“…We can now give a correct version of [3,Theorem 3.4], under the additional assumption of being left restriction.…”