Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

Abstract

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 HE(e)class is a group for every projection e ∈ E. This means that its corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite Ehresmann semigroups whose categories are EI is a pseudovariety and we show in the infinite case, that the collection of Ehresmann semigroups whose categories have endomorphism monoids each having one idempotent is a quasivariety. We prove that the simple modules of the semigroup algebra kS (over any field k) are induced Schützenberger modules of the maximal subgroups of S. 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-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.

Ehresmann theory and partition monoids

“…Cf. [65,84,85]. of all Brauer partitions is a submonoid (called the Brauer monoid ) if and only if X is finite [21].…”

“…Recent years have seen a number of important studies of representations of inverse semigroups, especially those of Steinberg [89][90][91][92], aspects of which have been extended to Ehresman semigroups by Stein [65,[84][85][86][87][88]. 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.…”

Ehresmann theory and partition monoids

“…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.…”

“…[21-23, 49, 58].• Does every (regular) right-restriction Ehresmann semigroup embed as a (2, 1, 1)-subalgebra of some P fd X ? Cf [30,59…”