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.

The (ordinary) quiver of an algebra A is a graph that contains information about the algebra's representations. We give a description of the quiver of C PTn, the algebra of the monoid of all partial functions on n elements. Our description uses an isomorphism between C PTn and the algebra of the epimorphism category, En, whose objects are the subsets of {1, . . . , n} and morphism are all total epimorphisms. This is an extension of a well known isomorphism of the algebra of ISn (the monoid of all partial injective maps on n elements) and the algebra of the groupoid of all bijections between subsets of an n-element set. The quiver of the category algebra is described using results of Margolis, Steinberg and Li on the quiver of EI-categories. We use the same technique to compute the quiver of other natural transformation monoids. We also show that the algebra C PTn has three blocks for n > 1 and we give a natural description of the descending Loewy series of C PTn in the category form.

A large body of evidence shows that when comparing non-symbolic numerosities, performance is influenced by irrelevant continuous magnitudes, such as total surface area, density, etc. In the current work, we ask whether the weights given to numerosity and continuous magnitudes are modulated by top-down and bottom-up factors. With that aim in mind, we asked adult participants to compare two groups of dots. To manipulate task demands, participants reported after every trial either (1) how accurate their response was (emphasizing accuracy) or (2) how fast their response was (emphasizing speed). To manipulate bottom-up factors, the stimuli were presented for 50 ms, 100 ms or 200 ms. Our results revealed (a) that the weights given to numerosity and continuous magnitude ratios were affected by the interaction of top-down and bottom-up manipulations and (b) that under some conditions, using numerosity ratio can reduce efficiency. Accordingly, we suggest that processing magnitudes is not rigid and static but a flexible and adaptive process that allows us to deal with the ever-changing demands of the environment. We also argue that there is not just one answer to the question ‘what do we process when we process magnitudes?’, and future studies should take this flexibility under consideration.

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

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