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We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let S be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra k S (over any field k ) are formed by inducing the simple modules of the maximal subgroups of S via the corresponding Schutzenberger module. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. As a natural example, we consider the monoid PT n of all partial functions on an n-element set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice $M_{k}$ (the unique lattice with $k+2$ elements and one anti-chain of length $k$) and want to find the maximal $k$ for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such $k$ is $2$ and there are cases where there is no maximum. No other intermediate option is possible., 14 pages

We prove that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is $n-1$ for all $n\geq 1$. This is also the global dimension of the complex algebra of the category of all epimorphisms between subsets of an $n$-set. In our proof we use standard homological methods as well as combinatorial techniques associated to the representation theory of the symmetric group. As part of the proof, we obtain a partial description of the Cartan matrix of these algebras., 27 pages

Cognitions play a central role in the maintenance of social anxiety disorder (SAD). The aim of the present study was to develop a brief version of the social thoughts and beliefs scale (STABS) that can be used in clinical trials and experience sampling studies in which multiple repeated measures (e.g., weekly or daily measurements) are utilized. Our sample (n = 361) included both individuals diagnosed with SAD (n = 108) and non-anxious controls (n = 253). We used item response theory analyses to examine items of the STABS and kept only items which differentiated between 4 levels of SAD-related cognitions between the 5th and 95th percentiles of scores. This strategy resulted in a brief, seven-item scale—the mini-STABS. We then compared the mini-STABS with the full-length, 21-item STABS, as well as examined convergent and divergent validity for the brief measure. Results indicated that the mini-STABS was highly correlated with the 21-item STABS, and demonstrated similar patterns of associations with convergent and divergent measures compared to the 21-item STABS. These findings suggest that the mini-STABS is a psychometrically sound brief version of the STABS that provides similar information using fewer items. Implications for assessment of SAD are discussed.

We give a new proof for the Littlewood-Richardson rule for the wreath product F≀Sn where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F≀Sn to F≀Sn+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F≀FIn where FIn is the category of all injective functions between subsets of an n-element set.

In this paper we study the representation theory of three monoids of partial functions on an $n$-set. The monoid of all order-preserving functions (i.e., functions satisfying $f(x)\leq f(y)$ if $x\leq y$) the monoid of all order-decreasing functions (i.e. functions satisfying $f(x)\leq x$) and their intersection (also known as the partial Catalan monoid). We use an isomorphism between the algebras of these monoids and the algebras of some corresponding locally trivial categories. We obtain an explicit description of a quiver presentation for each algebra. Moreover, we describe other invariants such as the Cartan matrix and the Loewy length., 36 pages. Revised based on referee suggestions

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 $\mathbb{C}PT_{n}$, the algebra of the monoid of all partial functions on $n$ elements. Our description uses an isomorphism between $\mathbb{C}PT_{n}$ and the algebra of the epimorphism category, $E_{n}$, whose objects are the subsets of $\{1,\ldots, n\}$ and morphism are all total epimorphisms. This is an extension of a well known isomorphism of the algebra of $IS_{n}$ (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 $\mathbb{C}PT_{n}$ has three blocks for $n>1$ and we give a natural description of the descending Loewy series of $\mathbb{C}PT_{n}$ in the category form., Added quiver computation of the partial Catalan monoid. Fixed minor mistakes

$E$-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 ., Comment: Added Erratum