Certain fundamental congruences on a regular semigroup

Howie, James; Lallement, G.

doi:10.1017/s2040618500035334

Cited by 56 publications

(49 citation statements)

References 18 publications

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“…Some of our results overlap with those proved by Yamada in [12] in a somewhat different guise; on the other hand, there is some overlapping with the results of Howie and Lallement [5], proved there by different methods. We make frequent use of several results established in the last mentioned paper, and add to the list of interesting connections among special congruences on a regular semigroup.…”

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confidence: 77%

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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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confidence: 77%

“…The proofs in [5] are different from ours. Several characterizations of the semigroups in Theorem 4.1 in terms of spined products were established by Yamada [12].…”

Section: \Jt:a-*(a*a5) (Aes)mentioning

confidence: 65%

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

Petrich

1973

Glasgow Math. J.

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“…The Green relations will be noted as usual, and for brevity, a semilattice of groups-congruence will be called a SG-congruence. That each of the above minimum congruences exists is explained in [5], and also noted there are some of the following relationships which will be useful here:…”

Section: Preliminary Results"mentioning

confidence: 99%

“…It was shown in [5] that ηΠσis the smallest congruence p such that S/p is a semilattice of groups and is unitary. Thus, in general, ξ is strictly contained in η Π σ.…”

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confidence: 99%

Certain congruences on orthodox semigroups

Mills¹

1976

Pacific J. Math.

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“…Proof. Howie and Lallement [2] have shown that J>* is the least semilattice congruence on 5. Hence (i) and (ii) are equivalent.…”

Section: Preliminariesmentioning

confidence: 99%

Completely semisimple inverse Δ-semigroups admitting principal series

Trotter¹,

Tamura²

1977

Pacific J. Math.

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A Δ-semigroup is a semigroup whose lattice of congruences is a chain with respect to inclusion. A completely semisimple inverse Δ-semigroup that admits a principal series is characterized here as a semigroup that results from a particular series of ideal extensions of Brandt semigroups by Brandt semigroups. A characterization is given of finite inverse Δ-semigroups in terms of groups, Brandt semigroups, and one to one partial transformations of sets. Introduction.A A-semigroup is a semigroup whose lattice of congruences is a chain with respect to inclusion. Schein [8] and Tamura [ 11] showed that a commutative Δ-semigroup is either a quasicyclic group A, or a commutative nil semigroup B with the divisibility chain condition, or A 0 , or B\ We study here the structure of completely semisimple inverse Δ-semigroups with principal series. Such semigroups will be characterized in terms of Δ-groups, idempotent properties, and ideal extensions of Brandt semigroups by Brandt semigroups.In [11] it was shown that the least semilattice congruence on a Δ-semigroup has at most two classes. We begin by characterizing completely semisimple inverse semigroups admitting principal series and having this property.In the final section we show that each finite inverse Δ-semigroup determines a set of structure data that involves groups, Brandt semigroups and one to one partial transformations of sets. Conversely the semigroup can be reconstructed from the structure data.

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Certain fundamental congruences on a regular semigroup

Cited by 56 publications

References 18 publications

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

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

Certain congruences on orthodox semigroups

Completely semisimple inverse Δ-semigroups admitting principal series

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