Largest subsemigroups of the full transformation monoid

In this paper we are concerned with the following question: for a semigroup S, what is the largest size of a subsemigroup T?S where T has a given property? The semigroups S that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups T that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups U where the least rank of an element in U is specified. Numerous examples are given.     

Homomorphisms and congruences on regular semigroups with associate inverse subsemigroups

An associate inverse subsemigroup of a regular semigroup S is a subsemigroup T of S containing a least associate x* of each x ∈ S, in relation to the natural partial order ≤. In [1] the authors describe the structure of regular semigroups with an associate inverse subsemigroup, satisfying two natural conditions. In this paper we describe all *-homomorphisms and all *-congruences on such semigroups.     

Automatic subsemigroups of free products

We consider the automaticity of subsemigroups of free products of semigroups, proving that subsemigroups of free products, with all generators having length greater than one in the free product, are automatic. As a corollary, we show that if S is a free product of semigroups that are either finite or free, then any finitely generated subsemigroup of S is automatic. In particular, any finitely generated subsemigroup of a free product of finite or monogenic semigroups is automatic.     

On generalized inverse transversals

Let S be a regular semigroup, S° an inverse subsemigroup of S.S° is called a generalized inverse transversal of S, if V（x）∩S°≠Ф. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases.     

Semilattices of bisimple orthodox semigroups

A structure theorem for bisimple orthodox semigroups was given by Clifford [2]. In this paper we determine all homomorphisms of a certain type from one bisimple orthodox semigroup into another, and apply the results to give a structure theorem for any semilattice of bisimple orthodox semigroups with identity in which the set of identity elements forms a subsemigroup. A special case of these results is indicated for bisimple left unipotent semigroups.     

Identities of semigroup rings over semilattices of completely (0-) simple semigroups

LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.     

Special subgroups of regular semigroups

Extending the notions of inverse transversal and associate subgroup, we consider a regular semigroup S with the property that there exists a subsemigroup T which contains, for each x∈S, a unique y such that both xy and yx are idempotent. Such a subsemigroup is necessarily a group which we call a special subgroup. Here, we investigate regular semigroups with this property. In particular, we determine when the subset of perfect elements is a subsemigroup and describe its structure in naturally arising situations.     

Associate inverse subsemigroups of regular semigroups

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x∈S, in relation to the natural partial order ≤ _S . We describe the structure of a regular semigroup with an associate inverse subsemigroup, satisfying two natural conditions. As a particular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element. Research supported by the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI.     

Fundamental semigroups having a band of idempotents

The construction by Hall of a fundamental orthodox semigroup W _B from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup S _B that plays the role of W _B for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S)=B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice. The Hall semigroup W _B is a subsemigroup of S _B , as are the (weakly) idempotent connected semigroups V _B and U _B . We show how the structure of S _B can be used to extract information relating to arbitrary weakly B-abundant semigroups with (C). This work was carried out during a visit to Lisbon of the second author funded by the London Mathematical Society and while the first author was a member of project POCTI/0143/2003 of CAUL financed by FCT and FEDER.     

Reversibility properties of semigroup actions on homogeneous spaces

This paper studies reversibility of subsemigroups acting on homogeneous spaces. The reversor set of a subsemigroup is defined and it is related to the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group with finite center. Let L be a subgroup of?G. Assume that S is a subsemigroup of G with intS???. The main result characterizes the reversibility of the S-action on G/L in terms of the actions of S and L on the flag manifolds of G.     

The structure of medial weakly U-abundant semigroups

We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent.     

Sets satisfying the Central Sets Theorem

Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem . We investigate here the class of semigroups that have a subset with zero Følner density which satisfies the conclusion of the Central Sets Theorem. We show that this class includes any direct sum of countably many finite abelian groups as well as any subsemigroup of (?,+) which contains ?. We also show that if S and T are in this class and either both are left cancellative or T has a left identity, then S×T is in this class. We also extend a theorem proved in (Beiglböck et al. in Topology Appl., to appear), which states that, if p is an idempotent in β? whose members have positive density, then every member of p satisfies the Central Sets Theorem. We show that this holds for all commutative semigroups. Finally, we provide a simple elementary proof of the fact that any commutative semigroup satisfies the Strong Følner Condition.     

On the subsemigroup complex of an aperiodic Brandt semigroup

We introduce the subsemigroup complex of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The results include alternative characterizations of faces and facets, asymptotical estimates on the number of facets, or establishing when the complex is pure or a matroid.     

A class of unary semigroups admitting a Rees matrix representation

In Billhardt et al. (Semigroup Forum 79:101–118, 2009) the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, first presented in Blyth et al. (Glasgow Math. J. 36:163–171, 1994). The main result of the present paper, Theorem 2.15, establishes that a regular semigroup S with an associate inverse subsemigroup S ^? satisfies three simple identities if and only if it is isomorphic to a generalised Rees matrix semigroup M(T;A,B;P), where T is a Clifford semigroup, A and B are bands, with common associate inverse subsemigroup E(T) satisfying the referred identities, and P is a sandwich matrix satisfying some natural conditions. If T is a group and A, B are left and right zero semigroups, respectively, then the structure described provides a usual Rees matrix semigroup with normalised sandwich matrix, thus generalising the Rees matrix representation for completely simple semigroups.