Introduction

In recent years, extensive study has been made by Munn [11], Yamada [14] and others concerning the structure of special classes of regular semigroups. But it seems that no systematic study has so far been made regarding the structure of regular semigroups in general. The main objective of this work is to make an attempt in this direction.

Preston’s representation theorem for regular semigroups (Theorem 3.21, [2]) shows that every regular semigroup can be faithfully represented as a direct sum of matrices over groups with zero, the groups being the Schutzenberger groups of the distinct $\mathcal{D}$-classes of the regular semigroup. The structure of the regular semigroup $S$, thus depends only on the family of maximal subgroups and certain partial transformations of $S$. Now, given an indexed family of groups, the question of constructing all regular semigroups having this family as the family of its maximal subgroups arises. (Solution to this constitutes an approach to the main problem.) Such a construction obviously needs some additional structure on the given family of groups. One such structure is the concept of a regular system. Every regular system uniquely determines a regular semigroup and conversely every regular semigroup is, to within isomorphism, of this type.

A possible alternative approach is to generalize the method used by Munn [11] to determine the structure of all bisimple inverse semigroups. Given a uniform semilattice $E$ he shows that the set of all fundamental bisimple inverse semigroups whose semilattice of idempotents is $E$, is precisely the set of all transitive subsemigroups of the semigroup of all isomorphisms of principal ideals of $E$. Bisimple inverse semigroups are then obtained as idempotent separating extensions of fundamental bisimple inverse semigroups. While attempting to generalize Munn’s method to regular semigroups, it becomes necessary to generalize (i) Howie’s theorem for inverse semigroups to regular semigroups and (ii) Munn’s result stated above so as to make it applicable to all fundamental regular semigroups.

In either approach, the set of idempotents of a regular semigroup plays a significant role, and it becomes necessary to characterize it abstractly. With this objective in mind, the concepts of biordered sets and bimorphisms are introduced in Chapter 2. These are generalizations of semilattices and homomorphisms of semilattices respectively. In Chapter 3, the concepts of chain transformations, partial $\omega$-isomorphisms of biordered sets and partially transitive sets of partial $\omega$-isomorphisms are introduced. Partial $\omega$-isomorphisms correspond to isomorphisms of principal ideals in Munn’s theory. Partially transitive sets of partial $\omega$-isomorphisms, which are generalizations of transitive semigroups in Munn’s theory, aid to determine all fundamental regular semigroups which are not necessarily bisimple.

Chapter 4 is a collection of results on regular semigroups needed in the sequel. The principal results are (i) the set of idempotents of a regular semigroup is a biordered set and (ii) Howie’s theorem for regular semigroups. (An alternate proof for (ii) is available in [5]). The result that the restriction of a homomorphism of a regular semigroup to its set of idempotents is a bimorphism, is obtained here.

In Chapter 5, the structure of fundamental regular semigroups is determined. Given a partially transitive set $\mathcal{T}$ on a biordered set $E$, an equivalence relation $\sigma$ on $\mathcal{T}$ and a binary operation on $\mathcal{T}∕\sigma$ are defined so as to make $\mathcal{T}∕\sigma =\overline{\mathcal{T}}$ a fundamental regular semigroup. Further, given a fundamental regular semigroup $S$, it is shown that there exists a partially transitive set ${\mathcal{T}}_S$ on the biordered set of idempotents $E\left(S\right)$ of $S$ such that the fundamental regular semigroup ${\overline{\mathcal{T}}}_S$ determined by ${\mathcal{T}}_S$ is isomorphic to $S$.

Since every regular semigroup is an idempotent separating extension of a fundamental regular semigroup, the latter result implies that the structure of all regular semigroups could be determined by Munn’s method referred to earlier. It is established that every biordered set is the set of idempotents of some regular semigroup. This result, together with (i) of Chapter 4, shows that a set with a structure which involves two quasi-orders is a biordered set if and only if it is the set of idempotents of a regular semigroup.

The principal objective in Chapter 6, is to characterize homomorphisms of fundamental regular semigroups in terms of bimorphisms and mappings of associated partially transitive sets. If $\mathcal{T}$ and $\mathcal{T}\prime$ are partially transitive sets on $E$ and $E\prime$ respectively and if $\theta$ is a bimorphism of $E$ into $E\prime$, then it is shown that there exists a homomorphism $h\left(\nu ,\theta \right)$ of $\bar{\mathcal{T}}$ into $\bar{\mathcal{T}}\prime$ that extends $\theta$ if and only if there exists a mapping $\nu$ of $\mathcal{T}$ into $\mathcal{T}\prime$ compatible with the given bimorphism $\theta$. It is also shown that $h\left(\nu ,\theta \right)$ is injective (surjective) if and only if $\nu$ is injective (surjective). In particular, given a bimorphism $\theta :E\to E\prime$ there exists a homomorphism $\bar{\theta }$ of ${\overline{Ẽ}}_0$ into ${\overline{Ẽ}}_0$ such that $\bar{\theta }$ is an isomorphism if and only if $\theta$ has this property. Again, a mapping of a biordered set into another is a bimorphism if and only if it extends to a homomorphism of some regular semigroup.

Given a regular semigroup $S$, Hall [5], recently gave different constructions for the greatest fundamental regular semigroup whose idempotent set is $E\left(S\right)$ and also for a fundamental regular semigroup isomorphic to $S∕\mu \left(S\right)$ where $\mu \left(S\right)$ is the greatest idempotent separating congruence on $S$.

The last two Chapters are devoted to the construction of regular semigroups according to the first approach. In Chapter 7, symmetric pre-sheaves of groups and regular family of matrices are defined. The concept of a pre-sheaf is as outlined in [1]. Regular groupoids are introduced as a first step towards the construction of regular semigroups. It is shown that a pair $\left(\mathcal{G},\mathcal{P}\right)$ consisting of a symmetric pre-sheaf $\mathcal{G}$ of groups over a biordered set $E$ and a regular family of matrices $\mathcal{P}$ for $\mathcal{G}$, determines to within isomorphism, a regular groupoid. It is also shown that a regular groupoid can be associated with a regular semigroup in a unique fashion. In Chapter 8, regular systems are defined and it is proved that every regular system $\mathcal{ℛ}$ determines a regular semigroup $\mathfrak{S}\left(\mathcal{ℛ}\right)$ and that given a regular semigroup $S$, there exists a regular systems ${\mathcal{ℛ}}_S$ such that $\mathfrak{S}\left({\mathcal{ℛ}}_S\right)$ is isomorphic to $S$. This correspondence between regular systems and regular semigroups is found to be essentially one-to-one. Since these results can be simplified considerably in the case of inverse semigroup, they are just mentioned.

In order to construct all regular semigroups according to the second approach, it is necessary to have idempotent separating extension theorems for regular semigroups. Results of this kind are already known for more restricted classes of semigroups, such as bisimple inverse semigroups and inverse semigroups (cf. Munn, [11] and D’Alarcao, [4]). Using methods similar to those described in this thesis, it is possible to obtain idempotent separating extension theorems for regular semigroups in general. These results are to be published elsewhere.

It may be noted that the structure of regular semigroups obtained here is in terms of the structure of biordered sets and groups. Thus, in order to understand the structure of regular semigroups completely, it is necessary to know more about the structure of biordered sets. The structure of biordered sets appears to be very complicated and it is not known whether similar structures have been studied elsewhere. However, it seems that a theory generalizing the theory of semilattices can be developed which will be of interest in itself.