Erratum to: Algebras of Ehresmann semigroups and categories

Abstract

Shoufeng Wang discovered an error in the main theorem of the author's Semigroup Forum article 'Algebras of Ehresmann semigroups and categories'. 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. A correction to the main theorem of [3]Shoufeng Wang [6] has observed that the proof of [3, Theorem 3.4] does not hold without the additional assumption that the semigroup is left restriction. 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].Communicated by Victoria Gould.The online version of the original article can be found under

Algebras of Reduced $E$-Fountain Semigroups and the Generalized Ample Identity

“…Steinberg's result is now fundamental in the study of representations of finite inverse semigroups. Guo and Chen [7] obtained a similar result for finite ample semigroups and the author extended this generalization to a class of right restriction E-Ehresmann semigroups [19,20] (E-Ehresmann semigroups were introduced by Lawson in [11]). This result has led to several applications regarding semigroups of partial functions [18,21,22,13] and recently also to the study of certain partition monoids [3].…”

“…In this case Corollary 5.3 is already known. The fact that ϕ is an isomorphism of k-algebras was proved by the author in[19,20]. The necessity of the right restriction property was proved by Wang in [26, Lemma 4.3].…”

Ehresmann theory and partition monoids

“…The main result of [79,80] concerns another connection between the E-Ehresmann semigroup S and the category C(S, E). To state it, we also need the orders ≤ r and ≤ l defined, respectively, on left and right E-Ehresmann semigroups:…”

“…Recent years have seen a number of important studies of representations of inverse semigroups, especially those of Steinberg [83][84][85][86], aspects of which have been extended to Ehresman semigroups by Stein [59,[78][79][80][81][82]. A crucial role in these studies is played by an isomorphism between the semigroup algebra of an appropriate Ehresmann semigroup and an associated category algebra, coming from the Ehresmann-Schein-Nambooripad/Lawson correspondence alluded to above.…”

Representation theory of order-related monoids of partial functions as locally trivial category algebras

“…In [25,Section 5] the author proved that each one of the monoid algebras k PO n , k PF n and k PC n is isomorphic to a category algebra of some corresponding subcategory of E n . These are actually examples of an isomorphism that holds for a larger class of semigroups (for details, see [24,26,31]). In [25,Section 5] this isomorphism was used in order to describe the ordinary quiver of these algebras.…”