Inverse Semi-Groups with Minimal Right Ideals

Preston, G. B.

doi:10.1112/jlms/s1-29.4.404

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“…The Preston-Vagner representation [16], [21] is a faithful representation for each inverse semigroup S. Hence, as an immediate consequence of Proposition 2.3, we have Theorem 2.4. Let S be an inverse semigroup.…”

Section: P Full 7-±±-»smentioning

confidence: 80%

Groups, semilattices and inverse semigroups. I, II

McAlister¹

1974

Trans. Amer. Math. Soc.

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ABSTRACT. An inverse semigroup S is called proper if the equations ea = e = e2 together imply a2 = a for each a, e E S. In this paper a construction is given for a large class of proper inverse semigroups in terms of groups and partially ordered sets; the semigroups in this class are called /"-semigroups. It is shown that every inverse semigroup divides a P-semigroup in the sense that it is the image, under an idempotent separating homomorphism, of a full subsemigroup of a P-semigroup. Explicit divisions of this type are given for u-bisimple semigroups, proper bisimple inverse semigroups, semilattices of groups and Brandt semigroups.The algebraic theory of inverse semigroups has been the subject of a considerable amount of investigation in recent years. Much of this research has been concerned with the problem of describing inverse semigroups in terms of their semilattice of idempotents and maximal subgroups. For example, Clifford [2] described the structure of all inverse semigroups with central idempotents (semilattices of groups) in these terms. Reilfy [17] A homomorphism 9 of an inverse semigroup S into an inverse semigroup T is called idempotent separating if 9 is one-to-one on the idempotents of S; that is, e2 = e,f2 = f, e9 = f9 imply e = /. A congruence is idempotent separating if it is the congruence of an idempotent separating homomorphism. Howie [5] showed that u = {(a,b) E S X S: aTxea = b~xeb for all e2 = e E S} is the maximum idempotent separating congruence on an inverse semigroup S. Further, Munn [13] has shown that T = S/p is fundamental; that is, the identity is the only congruence on T contained in <=¥. The importance of fundamental inverse semigroups stems from the fact (Munn [11], [13]) that, if T is a

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Section: P Full 7-±±-»smentioning

confidence: 80%

Groups, semilattices and inverse semigroups. I, II

McAlister¹

1974

Trans. Amer. Math. Soc.

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“…This corollary is an evident inference from Theorem 1 in [5]. (In [5] the term 'primitive', as applied to inverse semigroups, was used in a wider sense than here.) COROLLARY …”

Section: Primitive Regular Semigroupsmentioning

confidence: 93%

“…Thus to complete the proof of the lemma it only remains to show that every non-zero idempotent of Sjn is primitive. [5] Matrix representations of inverse semigroups 33…”

Section: Let S = S° Be a Regular Semigroup And P A O-restricted Congrmentioning

confidence: 99%

Matrix Representations of Inverse Semigroups

Preston

1969

J. Aust. Math. Soc.

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In his paper [1], W. D. Munn determines the irreducible matrix representations of an arbitrary inverse semigroup. Munn also gives a necessary and sufficient condition upon a 0-simple inverse semigroup for it to have a non-trivial matrix representation and for such semigroups gives a complete account of their representations. Munn's results rest upon the earlier work of Clifford [2] in which the representations of Brandt semigroups were determined. An alternative account of such representations was given by Munn in [3]. This earlier work is presented in Sections 5.2 and 5.4 of [4].In this paper we obtain a complete determination of the matrix representations of inverse semigroups. We restrict ourselves to inverse semigroups with a zero, and there is clearly no loss in generality in so doing. We show that any representation (without a null component) decomposes into what we term primitive components. Each primitive component in turn decomposes into representations which are determined by representations of certain associated Brandt semigroups. The set of Brandt semigroups involved is determined by what we call the representation ideal series of the representation.Conversely, representation ideal series are abstractly characterized and it is shown that starting with a representation ideal series and the Brandt semigroups it determines then representations of these Brandt semigroups determine, in a unique fashion, a representation of the original semigroup.The methods used are a development of those used by Munn in [1]. The results of Munn may be easily inferred from our results.The terminology and notation followed will be that of [4]. Certain differences from the terminology already used by Munn in earlier work are adopted for systematic reasons in conformity with [10]. Concepts and terminology not in [4] will be defined.The main theorem of this paper was announced in [13].

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“…By Lemma 1 of [4] and its left-right dual SeC\Sf = Sef and eSr\fS = efS. Hence since Se = eSr\Se and Sf=fSr\Sf, it follows that S,r\Sf = efSr\Sef=Sef.…”

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confidence: 94%

A Note on Representations of Inverse Semigroups

Preston¹

1957

Proceedings of the American Mathematical Society

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It is known [l; 2] that every inverse semigroup 5 has a faithful representation as a semigroup of (1, l)-mappings of subsets of a set A into A. The set A may be taken as the set of elements of 5 and the (1, l)-mappings as mappings of principal left ideals of 5 onto principal left ideals of 5. If £ is the set of idempotents of 5 then there is also a representation of 5, not necessarily faithful, as a semigroup of (1, l)-mappings of subsets of E into E [2]. If e££ denote by Se the subsemigroup eSe of 5. In this note we give a representation of any inverse semigroup S as a semigroup of isomorphisms between the semigroups Se. The representation is faithful if (a more general condition is given below) the center of each maximal subgroup of 5 is trivial.We recall that an inverse semigroup [3] is a semigroup S in which for any aES the equations xax -x and axa = a have a unique common solution xES called the inverse of a and denoted by a~l [5; 6]. This implies that the idempotents of 5 commute and that to each aES there corresponds a pair of idempotents e, f such that aa~l = e, a~la =/, ea = a, af=a. The idempotents e, f are called respectively the left and right units of a. For any two elements a, bES, (ab)~1 = b~ia~1 (see [3]). Throughout what follows 5 will denote an inverse semigroup and E will denote its set of idempotents.If e£E then S, will denote the subsemigroup eSe of 5. Lemma 1. If e,fEE then Ser\S/ = Sef.Proof. By Lemma 1 of [4] and its left-right dual SeC\Sf = Sef and eSr\fS = efS. Hence since Se = eSr\Se and Sf=fSr\Sf, it follows that S,r\Sf = efSr\Sef=Sef.

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Inverse Semi-Groups with Minimal Right Ideals

Cited by 18 publications

References 0 publications

Groups, semilattices and inverse semigroups. I, II

Groups, semilattices and inverse semigroups. I, II

Matrix Representations of Inverse Semigroups

A Note on Representations of Inverse Semigroups

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