Mario Petrich

Completely Regular Semigroups

A semigroup S is completely regular if and only if it is a disjoint union of groups. This concept, so simple in its formulation, has intrigued investigators for over forty years. In the early days, the underlying proposition was that the theory of such objects must necessarily be readily derivable from that for groups. The early work of Rees and Clifford gave some support to this notion. However, the work of recent years, especially that on varieties, has shown that the study of completely regular semigroups requires its own ingenious arsenal of tools.

Congruences on regular semigroups

Let S be a regular semigroup and let p be a congruence relation on S. The kernel of p, in notation ker p, is the union of the idempotent eclasses. The trace of p, in notation trp, is the restriction of p to the set of idempotents of S. The pair (kerp, trp) is said to be the congruence pair associated with p. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ((p V £)/£, ker p, (p V R)/R) is said to be the congruence triple associated with p. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S. For congruence relations p and 0, put pTz 0 [pTrr 0, pTo] if andonlyifpV: =0V: [pvR =0VR,trp=tr0]. ThenTz, Tr and T are complete congruences on the congruence lattice of S and T = Tl n T. Introduction and summary. After it was realized by Wagner that a congruence on an inverse semigroup S is uniquely determined by its idempotent classes, Preston provided an abstract characterization of such a family of subsets of S called the kernel normal system (see [2, Chapter 10]). This approach was the only usable means for handling congruences on inverse semigroups for two decades. A new approach to the problem of describing congruences on inverse semigroups was sparked by the work of Scheiblich [13] who described congruences in terms of kernels and traces. A systematic exposition of the achievements of this approach can be found in [10, Chapter III]. It was recognized by Feigenbaum [3] that every congruence p on a regular semigroup S is uniquely determined by its kernel, kerp, equal to elements tequivalent to idempotents, and its trace, trp, equal to the restriction of p to the set E(S) of idempotents of S. In the case of an inverse semigroup S, kerp and trp, as well as their mutual relationship, can be described abstractly by means of simple conditions on a subset of S and an equivalence on E(S) (see [10, Chapter III]). Following in the footsteps of Scheiblich, for orthordox and arbitrary regular semigroups, Feigenbaum [3] and lYotter [14] adopted the following approach: trp is characterized abstractly and to each such trp all matching kernels are described. This unbalances the symmetry of the kernel-trace approach by giving preference to the trace. Hence a balanced view relative to the kernel and the trace is evidently called for. The unqualified success of the kernel-trace approach for inverse semigroups, including its diverse ramifications, gave a certain hope that this may also turn out to be the case for regular semigroups. Judging by the complexity of regular semigroups and the attempts made for both orthodox and general regular semigroups, Received by the editors November 2, 1984. 1980 Mathematics Subject Classification. Primary 20M10; Secondary 08A30. (a)1986 American Mathematical Society 0002-9947/86

The structure of completely regular semigroups

1.00 +

The Congruence Lattice of a Regular Semigroup

.25 per page

A structure theorem for completely regular semigroups

The principal result is a construction of completely regular semigroups in terms of semilattices of Rees matrix semigroups and their translational hulls. The main body of the paper is occupied by considerations of various special cases based on the behavior of either Greens relations or idempotents. The influence of these special cases on the construction in question is studied in considerable detail. The restrictions imposed on Greens relations consist of the requirement that some of them be congruences, whereas the restrictions on idempotents include various covering conditions or the requirement that they form a subsemigroup.

Certain varieties and quasivarieties of completely regular semigroups

Let S be a regular semigroup and Con S the congruence lattice of S. If C is an isomorphism class of semigroups and ϱϵCon S, then we say that ϱ is over C if the idempotent ϱ-class belong to C. On Con S we can introduce the relations U, V, Tl, Tr and T as follows: if ϱ, θ, ϵConS, then we say that ϱ and θ are U− [V−, Tl−, Tr−, T−] related if both ϱ/ϱ∩θ and θ/ϱ∩gq over completely simple semigroups [rectangular band, left groups, right groups, groups]. It is shown that U, V, Tl, Tr and T are complete congruences on Con S.Various other characterizations of these congruences on Con S are obtained. Some of the congruences are studied for completely regular semigroups, orthodox semigroups and bands of groups. Further, since for any regular semigroup S, V∩Tl∩Tr is the identity relation, we obtain a subdirect decomposition of Con S.

Completely 0-simple semigroups of quotients III

Structure theorems for arbitrary completely regular semigroups have been given by Yamada, Warne, Petrich, and Clifford. A new structure theorem for these semigroups is proved here. It is reminiscent of both the structure theorem of Clifford-Petrich and of the Rees construction for completely simple semigroups. It simplicity ought to prove useful in the study of various aspects of completely regular semigroups. Completely regular semigroups, also called unions of groups, along with inverse semigroups, represents one of the most promising classes of semigroups from the point of view of construction, properties, congruences, varieties, amalgamation,etc. Their global structure in terms of semilattices of completely simple semigroups was discovered by Clifford in an early paper [1]. Their local structure is described by means of the Rees theorem for completely simple semigroups [5] which also stems from the same time period. This took care of the coarse structure of completely regular semigroups. A description of the interaction of the various completely simple components of a completely regular semigroup, that is its fine structure, had to wait quite a long time. After the structure of completely regular semigroups belonging to some restricted classes had been successfully treated, Yamada [7], Warne [6], Petrich [4], and Clifford-Petrich [2] came up with different constructions of a general completely regular semigroup. Unfortunately, all of these constructions are notationally quite complex. We offer here a simple construction of completely regular semigroups. Its rudiments can be traced to [2] and incorporates an idea of J. A. Gerhard. Its setting is reminiscent of the Rees construction for completely simple semigroups with ingredients which are both few in number and transparent in nature. The surprisingly simple form of this construction should prove useful in diverse studies of completely regular semigroups, such as the structure of special classes, free objects, etc. All necessary background, and in particular the notation and terminology, can be found in Petrich [3]. THEOREM. Let Y be a semilattice. For every a E Y, let SO = MJ(Ia, Go I Aa; PO) be a Rees matrix semigroup such that P. is normalized at an element also denoted Received by the editors November 25, 1985. 1980 Mathemotics Subject Clasification (1985 Revision). Primary 20M10.

Varieties of groups and of completely simple semigroups

This property is equivalent to 5 being a union of its (maximal) subgroups, and for this reason, these semigroups are frequently called unions of groups. Another equivalent definition is that they are semilattices of completely simple semigroups, a result due to Clifford [2], which is of fundamental importance for studying their structure, and hence they are occasionally referred to as Clifford semigroups. Further characterizations of these semigroups can be found in the books [3] and [14]. The class ^S% of completely regular semigroups does not form a variety, for it is not closed under taking subsemigroups. However, if we consider elements of ^

Operators related to idempotent generated and monoid completely regular semigroups

% as algebras with two operations (5, • , ~), where • is the given semigroup operation and x —> x 1 is the unary operation on S satisfying conditions (1), then ^ ^ ? constitutes a variety of universal algebras. Note that for x £ S and 5 in 9 ^ , x~ is the inverse of x in the maximal subgroup of 5 containing x. The purpose of this work is to study certains subvarieties and subquasivarieties of the variety ^S% of universal algebras just introduced. We first present two diagrams of the objects under study. The notation introduced in the two diagrams is fixed throughout the paper. Section 2 contains some special notation and terminology. We start with a study of subvarieties of orthodox bands of groups, normal bands of groups, and of orthodox normal bands of groups. The principal results in Sections 3-5 are the isomorphisms:

Regular semigroups satisfying certain conditions on idempotents and ideals

In a recent paper [6] the authors introduced the concept of a completely 0-simple semigroup of quotients. This definition has since been extended to the class of all semigroups giving a definition of semigroup of quotients which may be regarded as an analogue of the classical ring of quotients. When Q is a sernigroup of quotients of a semigroup S , we also say that S is an order in Q .