Note on idempotent semigroups, II

Yamada, Masashi; Kimura, Naoki

doi:10.3792/pja/1195524790

Cited by 51 publications

(26 citation statements)

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“…(1) for orthodox case, Fantham [5], Clifford [3]; (2) bands of groups, Clifford [2], Leech [10]; (3) normal bands of groups, the author [14]; (4) orthodox bands of groups, Yamada [16]; (5) bands, Kimura [8], Yamada and Kimura [18], the author [13]; (6) subdirect products of a band and a semilattice of groups, Yamada [16], [17], Howie and Lallement [6], the author [15]; (7) semilattices of groups, Clifford [1]. This list does not include numerous papers dealing with special properties of semigroups which have been successful only for certain classes of completely regular semigroups, particularly for semilattices of groups.…”

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confidence: 99%

The Structure of Completely Regular Semigroups

Petrich¹

1974

Transactions of the American Mathematical Society

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ABSTRACT. 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 Green's relations or idempotents. The influence of these special cases on the construction in question is studied in considerable detail. The restrictions imposed on Green's 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.1. Introduction and summary. The semigroups in the title are also known as "unions of groups", for they coincide with semigroups which are unions of their (maximal) subgroups, and "Clifford semigroups" because Clifford laid the groundwork for an eventual determination of their structure.Clifford [1] proved that these semigroups are semilattices of completely simple semigroups and conversely. Using Clifford's representation of a completely regular semigroup, Lallement [9] reduced the problem of their structure to the structure of completely simple semigroups and certain functions among them and their translational hulls, see Theorem 1 below. The structure of a completely simple semigroup is given in terms of Rees matrix semigroups by the Rees theorem, see [4, §3.2]. The author [12] gave two constructions of the translational hull of a Rees matrix semigroup; one of these is summarized in Theorem 2 below.A suitable combination of these results should then yield the structure of completely regular semigroups. This is precisely the underlying idea of the present work. However, the multiplication in the general case, Theorem 3 below, is very complicated and one cannot say that the structure of these semigroups is thus determined. Hence we are led to consider special cases; these are either given in terms of the behavior of idempotents or Green's relations, or the simplification of the construction of the general case.The structure of completely regular semigroups belonging to some special class has attracted wide attention. Properties or constructions of these were given by:

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The Structure of Completely Regular Semigroups

Petrich¹

1974

Transactions of the American Mathematical Society

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“…Thus to complete the picture, attention is focused on determining when such a primitive covering induces a partial skew lattice structure on the quasi-ordered set. The answer, given in Theorem 3.12 in terms of coset projections, directly generalizes the picture of normal bands given by Yamada and Kimura in [20]. The section concludes with a discussion of partial skew lattices for which coset bijections between comparable equivalence classes form a category under the standard composition of partial bijections.…”

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The Geometric Structure of Skew Lattices

Leech¹

1993

Transactions of the American Mathematical Society

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Abstract. A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric object. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Various aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators.Recall that a band is a semigroup satisfying the idempotent law: xx = x. Upon examining bands which are multiplicative subsemigroups of rings, one uncovers classes of bands which also possess an idempotent countermultiplication. This leads one to define a skew lattice to be an algebra with a pair of associative idempotent binary operations, the join and the meet, which are connected by a set of absorption laws (see 1.1). While skew lattices of idempotents in rings remain important sources of motivation, the results of [10-12] make it clear that skew lattices can sustain mathematical life on their own. Perhaps the most natural way to think about a skew lattice is as a noncommutative analogue of a lattice. As such, a skew lattice is not only an algebraic object, but also a geometric object. Thus far most of the research given in [10][11][12] has emphasized the algebraic side of skew lattices. The purpose of this paper is to investigate their geometric aspects and in particular the role of the natural partial order in determining their algebraic structure, much as a lattice is determined by its natural partial ordering. It is not our goal, however, to reinvent lattice theory. Hence the geometry of a skew lattice will be studied relative to the fixed structure of its underlying lattice. Saying this entails an implicit reference to the fundamental Clifford-McLean Theorem which in effect provides a first sketch of a skew lattice: a congruence is defined on each skew lattice (called natural equivalence) which induces its maximal lattice image and whose equiv-

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“…PROPOSITION Normal bands. Recall [14] that a band B is normal if it satisfies the identity axya = ayxa; B is left (resp. right) normal if it satisfies the identity axy = ayx (resp.…”

Section: Bs Sab = Sab 2 For All Abes (Iii) S Is Completely Regularmentioning

confidence: 99%

Regular semigroups which are subdirect products of a band and a semilattice of groups

Petrich

1973

Glasgow Math. J.

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1. Introduction and summary. In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, Jf may be a congruence, and soon.Instead of making some arbitrary choice of these conditions, we consider, in a certain sense, the converse problem, by starting with regular semigroups which are subdirect products of a band and a semilattice of groups. This choice, in turn, may seem arbitrary, but it is a natural one in view of the following abstract characterization of such semigroups: they are bands of groups and their idempotents form a subsemigroup. It is then quite natural to consider various special cases such as regular semigroups which are subdirect products of: a band and a group, a rectangular band and a semilattice of groups, etc. For each of these special cases, we establish (a) an abstract characterization in terms of completely regular semigroups satisfying some additional restrictions, (b) a construction from the component semigroups in the subdirect product, (c) an isomorphism theorem in terms of the representation in (b), (d) a relationship among the congruences on an arbitrary regular semigroup which yield a quotient semigroup of the type under study.Using the following abbreviations: B-bands; SG-semilattices of groups; L-left zero semigroups; R-right zero semigroups; S-semilattices; G-groups; 1-one element semigroups, we consider regular semigroup subdirect products of these according to the diagram below.This leads us to classes of semigroups for which we introduce the following abbreviations: V-a variety of bands; UVG-semigroups 5 for which $e is a congruence, S/Jfe V, and the idempotents form a unitary subset of S; M-rectangular bands; CRISN-completely regular semigroups whose idempotents form a strongly normal subband (defined in §2); CRILSNare CRISN with left added in front of" strongly "; CRUSN-are CRISN whose idempotents form a unitary subset; CRTJLSN-conjunction of the last two. In addition we adopt from [5]: ISBG-bands of groups whose idempotents form a subsemigroup. Thus, we call a congruence p on a semigroup S an A'-congruence if Sjp is in class X, e.g., for X = G, a group congruence, for X = S, a semilattice congruence (should not be confused with the letter S which usually denotes a semigroup).

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Note on idempotent semigroups, II

Cited by 51 publications

References 1 publication

The Structure of Completely Regular Semigroups

The Structure of Completely Regular Semigroups

The Geometric Structure of Skew Lattices

Regular semigroups which are subdirect products of a band and a semilattice of groups

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