Showing papers on "Bicyclic semigroup published in 1981"
55 citations
38 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].
14 citations
10 citations
01 Feb 1981
TL;DR: In this article, the syntactic monoid syn(H) is characterized as a monoid with a disjunctive,A-zero, and the two particular interesting cases when synH is a nil monoid and when syn H is a semillatice are also characterized.
Abstract: If X* is the free monoid generated by the alphabet X, then any subset L of X* is called a language over X. If PL is the principal congruence determined by L, then the quotient monoid syn(L) = X*/PL is called the syntactic monoid of L. A hypercode over X is any set of nonemtpy words that are noncomparable with respect to the embedding order of X*. If H is a hypercode, then the language H = {xlx E X* and a < x for some a E H) is a right convex ideal of X*. The syntactic monoid syn(H) can be characterized as a monoid with a disjunctive ,A-zero. The two particular interesting cases when syn(H) is a nil monoid and when syn(H) is a semillatice are also characterized.
9 citations
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.
6 citations
5 citations
3 citations
TL;DR: In this article, the semigroup of binary relations Bx for a finite set X is studied, and the ideal structure of Bx, the congruences on Bx and the endomorphisms are investigated.
Abstract: 0. Introduction. In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relations Bx for a finite set X: what is the ideal structure of Bx, what are the congruences on Bx, what are the endomorphisms of Bx? For |X| = n it is convenient to regard Bx as the semigroup Bn of n X n (0, l)-matrices under Boolean matrix multiplication. For all semigroup notation, terms, and facts used here without explanation, see [2].
2 citations
01 Jan 1981
TL;DR: In this article, a semigroup S is said to be finitely convergent if for every x in S, n n 1Sx'S = Sx mS for some positive integer m.
Abstract: According to Tamura [3], a semigroup S is said to be finitely convergent, if for every x in S, n n 1Sx'S = Sx mS for some positive integer m. Simple semigroups, bands, nil semigroups, finite semigroups, right regular semigroups, left regular semigroups, and intraregular semigroups are finitely convergent semigroups. The bicyclic semigroup is an example of a regular semigroup which is finitely convergent. A free inverse semigroup with one generator is an example of a regular semigroup which is not finitely convergent. This example was furnished by Maria Szendrei in one of the seminar meetings in Szeged, while I was visiting Hungary. However we have
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).
4.
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.