Ehresmann semigroups whose categories are EI and their representation theory
Introduction
A semigroup S is called inverse if every element has a unique inverse, that is, an element such that and . Inverse semigroups are one of the central objects of study in semigroup theory (see [20]) and in particular their representation theory is well-studied (see [40, Part IV]). There are several generalizations of inverse semigroups that keep some of their properties and structure. In this paper we discuss representations of a generalization called Ehresmann semigroups. Let E be a subsemilattice of a semigroup S (that is, a commutative subsemigroup of idempotents). Define two equivalence relations and in the following way: Elements satisfy () if a and b have the same set of right (respectively, left) identities from E. We also define . Assume that every and class contains precisely one idempotent from E denoted and respectively. The semigroup S is called Ehresmann (or E-Ehresmann if the set E is to be emphasized) if is a right congruence and is a left congruence. An Ehresmann semigroup is called right restriction if the identity holds for every and . The main class of semigroups we consider in this paper are finite right restriction Ehresmann semigroups with the additional property that the -class of every is a group. We call such semigroups right restriction EI-Ehresmann because the corresponding Ehresmann category is an EI-category (that is, every endomorphism is an isomorphism). This is a generalization of a class known in the literature as (finite) weakly ample semigroups (see [9, Section 4]). Natural examples of such semigroups are the monoid of all partial functions on the set and the monoid of full domain partitions on a finite set introduced in [6]. In Section 3 we prove that this class is a pseudovariety in signature . We also discuss cases when such semigroups are embeddable in as a bi-unary semigroup.
In Section 4 we show that the simple modules of the semigroup algebra (over any field ) of a finite right restriction EI-Ehresmann semigroup are given by induced Schützenberger modules of the simple modules of the maximal subgroups of S. Unlike inverse semigroups, Ehresmann semigroups algebras need not be semisimple, even over fields of “good” characteristic. Therefore, an Ehresmann semigroup can have non-semisimple projective modules. Another goal of this paper is to describe the indecomposable projective modules of finite right restriction EI-Ehresmann semigroups. For this we need a construction that replaces Green's -classes by -classes as the basis of induction. Let S be a finite semigroup and let be a subset of idempotents. Choose an and let be its -class. In Section 5 we characterize certain cases where (the maximal subgroup with unit element e) acts on the right of . In particular we prove that acts on the right of for every Ehresmann semigroup. If we denote by the -vector space with basis , this implies that is a -module for every -module V.
A very successful way to study algebras of inverse semigroups or their generalizations is to relate them to algebras of an associated category (or a “partial semigroup”). This is often done with an appropriate Möbius function as in [11], [12], [17], [31], [33], [38], [39], [42]. In particular, the second author has proved in [34], [35] the following theorem. Let S be a finite right restriction Ehresmann semigroup. Then the semigroup algebra (over any field ) is isomorphic to the category algebra for the associated Ehresmann category . This result has led to several applications in the representation theory of monoids of partial functions [36], [37]. Now assume again that S is a finite right restriction EI-Ehresmann semigroup and also assume that the order of every subgroup of S is invertible in . In Section 6 we use the isomorphism between the semigroup algebra and the corresponding category algebra mentioned above to prove that any module of the form (where V is a simple -module) is an indecomposable projective module of . Moreover, any indecomposable projective module is of this form. This gives a purely semigroup theoretic construction and it is very similar to other known constructions in the representation theory of semigroups [8], [26]. We also give a formula for the dimension of this module over an algebraically closed field. As an application, we consider the monoid in the case . It was already proved that the simple and indecomposable injective modules of can be described using induced and co-induced representations respectively (essentially this is a consequence of [24, Theorem 4.4]). In Section 7 we complete this picture by giving a description of the indecomposable projective modules of along with the natural epimorphism onto their simple images. The general formula for the dimension of the indecomposable projective modules boils down in this case to a combinatorial formula that sums up certain Kostka numbers. We also draw some conclusions regarding the Cartan matrix of by showing that certain entries of it are zero. Finally, we give a counter-example to the above isomorphism between the algebras of an Ehresmann semigroup and the associated category in the case where S is not right (or left) restriction hence showing that this requirement cannot be omitted. We also give an example of an EI-Ehresmann semigroup that is neither left nor right restriction but its algebra is isomorphic to the algebra of the corresponding Ehresmann category. Proofs of these final assertions can be found in the ArXiv version of this paper [23].
Section snippets
Semigroups
Let S be a semigroup and let be the monoid formed by adjoining a formal unit element. We denote by , , and the usual Green's equivalence relations: and . In a finite semigroup, it is known that , which is Green's relation . We denote by the set of all idempotents of S. It is well-known that if then its -class forms a group that we will denote by . This is the maximal subgroup of S with unit element e. An element
Some classes of Ehresmann semigroups
By an endomorphism of a category we mean a morphism f whose domain and range are the same object. For an object c of we call , the endomorphism monoid at c. Definition 3.1 A category is called an EI-category if every endomorphism is an isomorphism, that is, the endomorphism monoids are groups. Definition 3.2 Let S be an Ehresmann semigroup. We call S an EI-Ehresmann semigroup if
Simple modules of finite right restriction EI-Ehresmann semigroups
From now on the focus will be on the case of finite right restriction EI-Ehresmann semigroups. In this section we would like to describe the simple modules of algebras of such semigroups and in certain cases also the indecomposable injective ones. We remark that certain observations on the semisimple image of such semigroup algebras and their ordinary quiver can be found in [34, Proposition 5.17] and [25, Section 6.3]. Lemma 4.1 Let S be a finite right restriction EI-Ehresmann semigroup and let . Then
Construction of - modules
Our next goal is to obtain a description of indecomposable projective modules of finite right restriction EI-Ehresmann semigroups. The description involves an induction using the relation instead of . In this intermediate section we consider several cases where such a construction yields a well defined -module structure.
Let S be a fixed semigroup and let be a subset of idempotents. Lemma 5.1 Let and denote by the -class of a. Then S acts by partial functions on according to
Projective modules of right restriction EI-Ehresmann semigroups
In this section we study projective modules of finite Ehresmann and right restriction semigroups. The main result will be in the case of right restriction EI-Ehresmann semigroups. We start with some lemmas that will be of later use.
Lemma 6.1 Let S be an Ehresmann and right restriction semigroup and let . Then, (where ≤ is the natural partial order on the subsemilattice E). Proof First note that If then by the right ample identity and by the right
The monoid of partial functions
Recall that denotes the monoid of all partial functions on the set . In this section we apply the results of Section 6 in this specific case. For this section we fix , the field of complex numbers. Let and denote by the partial identity function on A. As already mentioned in Section 3.2, is a right restriction EI-Ehresmann semigroup with as a subsemilattice of projections. Recall that we are composing functions from right to left. We denote the
Acknowledgements
The authors thank the referee for hisher helpful comments. We thank Professor Michael Kinyon and Professor Victoria Gould for a number of useful discussions and for the specific results we noted in the body of the paper. We thank Professor Timothy Stokes for pointing our attention to references [15] and [30] and for discussions related to their content and relation to this paper.
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