# Showing papers in "Semigroup Forum in 1981"

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TL;DR: In this paper, the authors introduced the terms Maximal Γ-ideal, primary Γsemigroup and prime ǫ-ideals, and proved that if S is a Γ semigroup with identity and if ( non zero, assume this if S has zero) proper prime Γ -ideals in S are maximal then S is primary Β-semigroup.

Abstract: In this paper, the terms, Maximal Γ-ideal, primary Γ-semigroup and Prime Γ-ideal are introduced. It is proved that if S is a Γsemigroup with identity and if ( non zero, assume this if S has zero) proper prime Γ-ideals in S are maximal then S is primary Γ-semigroup. Also it is proved that if S is a right cancellative quasi commutative Γ-emigroup and if S is a primary Γsemigroup or a Γsemigroup in which semiprimary Γideals are primary, then for any primary Γ-ideal Q, √ Q is non-maximal implies Q = √ Q is prime. It is proved that if S is a right cancellative quasi commutative Γ-semigroup with identity, then 1) Proper prime Γ-ideals in S are maximal. 2) S is a primary Γ-semigroup. 3) Semiprimary Γ-ideals in S are primary, 4) If x and y are not units in S, then there exists natural numbers n and m such that (x Γ) n-1 x = yΓs and (yΓ) m-1 y = xΓr. For some s, r ∈ S are equivalent. Also it is proved that if S is a duo Γ–semigroup with identity, then 1) Proper prime Γ– ideals in S are maximal. 2) S is either a Γ– group and so Archimedian or S has a unique prime Γ–ideal P such that S = G∪ P, where G is the Γ–group of units in S and P is an Archimedian sub Γ–semi group of S are equivalent. In either case S is a primary Γ–semigroup and S has atmost one idempotent different from identity. It is proved that if S is a duo Γ-semigroup without identity, then S is a primary Γ-semigroup in which proper prime Γ-ideals are maximal if and only if S is an Archimedian Γ-semigroup. It is also proved that if S is a quasi commutative Γ-semigroup containing cancellable elements, then 1) The proper prime Γ-ideals in S are maximal. 2) S is a Γ-group or S is a cancellative Archimedian Γ-semigroup not containing identity or S is an extension of an Archimedian Γ-semigroup by a Γ-group S containing an identity are equivalent. Mathematical subject classification (2010): 20M07; 20M11; 20M12.

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TL;DR: Finite Thue systems that are Church-Rosser and properties of the semigroups such systems present are studied in this article, where the properties of such systems are investigated.

Abstract: Finite Thue systems that are Church-Rosser and properties of the semigroups such systems present are studied.

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16 citations

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TL;DR: In this paper, the syntactic algebra of a code is defined as the algebra of the sub-monoid of the code generated by the code's generator, which is a special class of languages: free submonoids of the free monoid.

Abstract: {. INTRODUCTION There exists a canonical way to associate to each language (that is, to each subset of a free monoid) an algebra on a ~iven field; we call it its syntactic algebra. This construction is in fact a particular case of a more general one that applies to formal power series in non commutative variables similar to the construction of the syntactic monoid of a language. In this paper we are interested in syntactic algebras of a special class of languages : free submonoids of the free monoid. Recall that a submonoid of a free monoid is not necessarily free, and that if it is, then it admits a unique basis : such a basis is also called a code. A code is said to be prefix if it contains no left factor of any of its elements (this condition ensures that it is a code) and biprefix if the dual condition is also fullfilled. We call (by a slight abuse of language) syntactic algebra of a code the syntactic algebra of the submonoid it generates. When the code is

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TL;DR: In this paper, the authors considered the general case of varieties of inflations of unions of groups, obtaining a description of those identities which determine such varieties. And they also deduced Tamura's result as a special case and also answer several questions posed in [6].

Abstract: Tamura [6] has shown that a semigroup S satisfies an identity xy = f(x,y), where f(x,y) is a word that begins in y, ends in x and has length at least 3, if and only if S is an inflation of a semilattice of groups satisfying the same identity. We consider here the general case of varieties of inflations of unions of groups, obtaining a description of those identities which determine such varieties. We are able to deduce Tamura's result as a special case, and we also answer several questions posed in [6].

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TL;DR: In this article, the authors give a short alternative description of all regular four-spiral semigroups and their maximum completely simple homomorphic images in terms of bisimple ω-semigroups (whose structure is known by Reilly's theorem).

Abstract: A regular semigroup S is called an ℋ-coextension of a regular semigroup T if there exists an idempotent-separating homomorhism from S onto T. J. Meakin [5] has described all regular four-spiral semigroups, i.e. all ℋ-coextensions of the fundamental four-spiral semigroup Sp4 [2], by means of the structure mappings on a regular semigroup. The purpose of this note is to point out that D. Allen's generalization [1] of the Rees theorem allows one to give a short alternative description of all regular four-spiral semigroups and their maximum completely simple homomorphic images in terms of bisimple ω-semigroups (whose structure is known by Reilly's theorem [7]) and Rees matrix semigroups ℳ(S;I;ΛP) over a semigroup S [3]. The notion of a Rees matrix semigroup over a semigroup is also used to embed semigroups in idempotent-generated ones, providing easy proofs for some embedding theorems of F. Pastijn [6].

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TL;DR: In this article, the authors introduced the notion of group ideal in a semigroup and proved that all group ideals of a compact affine semigroup are convex and dense, which generalizes many results in the literature concerning ideals in semigroups.

Abstract: In this paper we introduce the notion of a group ideal in a semigroup. We shall prove that all group ideals of a compact affine semigroup are convex and dense. This generalizes many results in the literature concerning ideals in semigroups.

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TL;DR: In this article, the authors give further periodic properties of groupbound semlgroups, including periodic conditions on principal left (resp. right) ideals, denoted by PLs.

Abstract: An element a of a semigroup S is sa~d to be groupboun d [5] if some power of a belongs to a subgroup of S. The semigroup S is said to be groupbound if all its elements are groupbound. In [6] the authors gave some results that indicated the property of being groupbound was a kind of generalv ized periodicity. (L.N. Sevrin has called groupbound semigroups by quasiperiodic semigroups, [12, 13, 14, 15]. Earlier literature [8] has used "pseudoinvertible ~ instead of "groupbound.") In this paper, we give further periodic properties of groupbound semlgroups. The first section gives some results essentially from the literature. Most of these are for regular semigroups. In the second section two decomposition theorems, known [7] for periodic semigroups, are proved for groupbound semlgroups. We will follow standard notations as in Clifford and Preston [2], and Petrlch [I0]. I. GENERALITIES AND REGULARITY A sem~group S is said to have periodic condition on principal left (resp. right) ideals, denoted PL