The global dimension of the algebra of the monoid of all partial functions on an n-set as the algebra of the EI-category of epimorphisms between subsets

Ehresmann theory and partition monoids

“…The inverse isomorphism involves the Möbius function of the relevant order, but we do not need the details here. Theorem 2.6 allows the representation theory of the semigroup S to be reduced to that of the category C, and in many cases this is a substantial simplification; several examples are given in [79, Section 5.1] and also [78,81,83,84].…”

“…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.…”

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

“…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]. We mention also that Wang [26] generalized the above results further to a certain class of right P -restriction, P -Ehresmann semigroups (for definitions of these notions see [10]) -but we do not follow this approach in this paper.…”

Classifying global dimensions of category algebras of some finite combinational categories

“…In this paper we study representation theory of some combinatorial categories, an active area attracting the attention of a lot people because of its numerous applications in other areas such as representation stability theory; see for instances [1,2,3,4,5,7,8,10,11,12,14,15,16]. Specifically, since the category algebras (over a field) of many infinite combinatorial categories have infinite global dimension; see for instances [8,Thorem 4.9] for F I, the category of finite sets and injections, [5,Corollary 1.6] and [10, Theorem 1.9] for V I, the category of finite dimensional vector spaces over finite fields and linear injections, and [13,Corollary 5.18] for F I d , the category of finite sets and pairs of injections and d-coloring maps, we are motivated to establish the finite version of this result.…”

“…We briefly describe the essential idea to establish the above classification, which is different from the combinatorial representation theory approach described in [15,16], where Stein and Steinberg classified the global dimensions of some important finite monoids (over the complex field). Firstly, since these finite combinatorial categories are equivalent to truncations of corresponded infinite combinatorial categories respectively, we may view a representation a finite combinatorial category as a representation of the corresponded infinite combinatorial category via the lift process.…”