On decompositions of a commutative semigroup

Tamura, Takayuki; Kimura, Naoki

doi:10.2996/kmj/1138843534

Cited by 77 publications

(24 citation statements)

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“…In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9,10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by a, P and t] respectively.…”

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confidence: 99%

Certain fundamental congruences on a regular semigroup

Howie

Lallement

1966

Proceedings of the Glasgow Mathematical Association

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In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9,10] [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by a, P and t] respectively.In § 1 we establish connections between /? and t\ on the one hand and the equivalence relations of Green [7] (see also Clifford and Preston [4, § 2.1]) on the other. If for any relation H on a semigroup S we denote by K* the congruence on S generated by H, then, in the usual notation, In § 2 we show that the intersection of a with jS is the smallest congruence p on S for which Sip is a UBG-semigroup, that is, a band of groups [4, p. 26] in which the idempotents form a unitary subsemigroup. The structure of such semigroups (and indeed of semigroups more general than this) has been investigated by Fantham [6]; his theorem (or rather the special case that is of interest here) is described below. A corollary of our result is that a n rj is the smallest congruence p for which Sjp is a USG-semigroup, that is, a semilattice of groups with a unitary subsemigroup of idempotents.These results lead naturally to a study of RU-semigroups (regular semigroups whose idempotents form a unitary subsemigroup), lSBG-semigroups (bands of groups whose idempotents form a subsemigroup) and SG-semigroups (semilattices of groups), and to the consideration of the minimum RU-congruence K, the minimum ISBG-congruence ( and the minimum SG-congruence ^ on a regular semigroup. The principal results of § 3 are that en/? = £ v K and a n tj = /; v K.In § 4 we show that any UBG-congruence on a regular semigroup can be expressed in a unique way as T ny, where x is a group congruence and y is a band congruence. A similar result holds for USG-congruences.

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confidence: 99%

Certain fundamental congruences on a regular semigroup

Howie

Lallement

1966

Proceedings of the Glasgow Mathematical Association

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“…Let | be the finest S-congruence on S. Throughout this paper, we let fi = ß(S) = 5/| denote the maximal semilattice image of S. By a theorem of Tamura [18], [19], each component of £ is S-indecomposable and is called the S -indecomposable component of S. An ideal P of S is prime if S \ P is a subsemigroup of 5. In such a case {/*, S \ P) is a semilattice decomposition of S. If 5 is a commutative algebraic semigroup, then it follows from Corollary 1.4 below, and Tamura and Kimura [20] that E(S) = ß(S).…”

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confidence: 99%

On Linear Algebraic Semigroups

Putcha¹

1980

Transactions of the American Mathematical Society

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Abstract.Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of K " along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals /",...,/, such that S = /, 3 • • • D I0, I0 is the completely simple kernel of S and each Rees factor semigroup 4//*_i is either nil or completely 0-simple (k =■ 1,...,/). We say that S is connected if the underlying set is irreducible. We prove the following theorems (among others) for a connected algebraic semigroup S with idempotent set E(S).

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“…For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences p on a completely regular semigroup S such that S/p is a semilattice of groups.…”

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Certain congruences on a completely regular semigroup

Pirnot

1974

Glasgow Math. J.

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1. Introduction and summary. Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences p on a completely regular semigroup S such that S/p is a semilattice of groups. We shall call such a congruence an SGcongruence.Since Clifford has shown in [1] that a regular semigroup S is a semilattice of groups if and only if idempotents of S lie in the centre of S, it follows that the problem of rinding group congruences and commutative congruences on a completely regular semigroup S can be solved provided that one can describe the SG-congruences on S.§2 contains definitions and notation used in this paper. In §3 we determine all group congruences on a Rees matrix semigroup and use this in § §4 and 5 to find the smallest SGcongruence, the smallest commutative congruence and the smallest group congruence on a completely regular semigroup.The reader is referred to [2], [5] and [7] for terminology and concepts not presented in this paper.

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On decompositions of a commutative semigroup

Cited by 77 publications

References 2 publications

Certain fundamental congruences on a regular semigroup

Certain fundamental congruences on a regular semigroup

On Linear Algebraic Semigroups

Certain congruences on a completely regular semigroup

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