Showing papers on "Cancellative semigroup published in 1981"
55 citations
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 paper, a necessary and sufficient condi-tion for an element to be a divisor of zero in a semigroup ring is given, where G is an u.p. semigroup.
Abstract: In [2] Coleman and Enochs obtained results about the units of the polynomial ring R[x] for rings R satisfying a condi-tion which is, in some sense, a generalization of commutativity. In [3] some of these results were extended to group rings over an ordered group. In this note a class of rings larger than the class considered in [2] is used to extend the results in [2] and 3] to the semigroup ring RG, G an u.p, semigroup. In the last section we give a necessary and sufficient condi-tion for an element to be a divisor of zero in RG where G is an u.p. semigroup.
TL;DR: In this article, the existence of Θ-closeness of semigroup systems was studied in terms of commutative diagrams and necessary and sufficient conditions for solving the problem of monitoring the behavior of a semigroup, system.
Abstract: The main results of the present article are as follows.
1.
The introduction of tolerance Θ into set Yo induces equivalence relations into sets X, Y, and ϕ of semigroup system S such that there is a morphism of the initial system S into the semigroup factor-system Theorem 1).
2.
With the initial tolerance Θ is linked the concept of Θ-equivalence of systems, which is a generalization of the usual equivalence of systems. Theorem 2 on the factoring of the set of Θ-equivalent semigroup systems sia generalization of the familiar theorem on factoring of the set of equivalent semigroup systems.
3.
With the aid of the concept of Θ-closeness of semigroup systems we state necessary and sufficient conditions for solving the problem of monitoring the behavior of a semigroup, system (see Theorem 3).
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We state, in terms of commutative diagrams, the conditions for the existence of Θ-close semigroup systems. The connection is stablished between the Θ-closeness of a system to the initial semigroup system and the croperties of the covering induced by the Θ-close system into the set of states of the initial system (see Theorem 4).
5.
We give a procedure for constructing the Θ-model of a semigroup system which is formal rather than constructive. But it can be constructive for certain classes of semigroup systems. For instance, we showed ia [11] that the procedure can be used for constructing Θ-models in the class of finite determinate automata. The procedure may also be used for the discrete approximation of continuous dynamic systems.
TL;DR: In this paper, it was shown that the existence of a semigroup of a left invariant measure satisfying certain conditions is equivalent to the imbeddability of some open ideal of the given semigroup as an open subsemigroup in a locally compact group of left fractions.
Abstract: In this paper the case of topological semigroups that are cancellative and right reversible is considered. It is shown (Theorem 1) that the existence in such a semigroup of a left invariant measure satisfying certain conditions is equivalent to the imbeddability of some open ideal of the given semigroup as an open subsemigroup in a locally compact group of left fractions. Conditions for imbedding topological semigroups in locally compact groups are considered in measure-theoretic terms. Bibliography: 8 titles.