Showing papers on "Bicyclic semigroup published in 1974"
167 citations
45 citations
TL;DR: A semigroup S (with zero) is called a right inverse semigroup if every (nonnull) principal left ideal of S has a unique idempotent generator as discussed by the authors.
39 citations
34 citations
31 citations
TL;DR: In this paper, the authors characterize Archimedean semigroups with idempotents satisfying (xy)m = xmym as exactly those semiggroups which are a retract extension of a completely simple semigroup by a nil semigroup.
Abstract: In this paper we characterize Archimedean semigroups with idempotents satisfying (xy)m = xmym as exactly those semigroups which are a retract extension of a completely simple semigroup satisfying (xy)m = xmym by a nil semigroup satisfying (xy)m = xmym. Regular semigroups satisfying (xy)2 = x2y2 are exactly those semigroups which are a spined product of a band and a semigroup which is a semilattice of Abelian groups. A semigroup which is a nil extension of a regular semigroup satisfies (xy)2 = x2y2 if and only if it is a retract extension of a regular semigroup satisfying (xy)2 = x2y2 by a nil semigroup satisfying (xy)2 = x2y2
20 citations
TL;DR: In this paper, the concepts of rectangular group and Bruck-Reilly construction were extended to produce some semigroups with midunits, and an abstract characterization was made of certain of these semiigroups.
Abstract: A semigroup S has a midunit u if aub=ab for all a and b in S. It is the purpose of this paper to explore semigroups containing midunits and to find an analog to the group of units. The concepts of rectangular group and Bruck-Reilly construction will be extended to produce some semigroups with midunits, and an abstract characterization will be made of certain of these semigroups.
15 citations
TL;DR: In this paper, the authors investigated under what conditions an inverse semigroup M is isomorphic to the syntactic monoid M(A) of afinite prefix code A over an alphabet X. They found a necessary condition for this to happen, which expresses a precise link between the group of units of M and the maximal subgroups of the 0-minimal ideal of M (Theorem 2.1).
Abstract: The purpose of this paper is to investigate under what conditions an inverse semigroup M is isomorphic to the syntactic monoid M(A)* of afinite prefix code A over an alphabet X. We find a necessary condition for this to happen. It expresses a precise link between the group of units of M and the maximal subgroups of the 0-minimal ideal of M (Theorem 2.1). The condition is shown to be sufficient in case M is an ideal extension of a Brandt semigroup by a group (Corollary 2.3). We also introduce and study stable codes (products of subsets of the alphabet) and give structural properties of their syntactic monoids (Proposition 3.3 and Theorem 3.5). Most of our results inter-relate structural properties of certain semigroups and divisibility of integers attached to them. The terminology follows [1] and [3].
11 citations
11 citations
TL;DR: This chapter shows that regardless of how a semigroups S is faithfully represented as a semigroup of partial functions on a set Q, the complexity of the resulting transformation semigroup is the same.
Abstract: Publisher Summary This chapter focuses on the complexity of semigroups and morphism. It shows that regardless of how a semigroup S is faithfully represented as a semigroup of partial functions on a set Q , the complexity of the resulting transformation semigroup is the same. As per the chapter, complexity can be regarded as a theory of semigroups rather than transformation semigroups. However, transformation semigroups will be used extensively as tools in complexity theory.
10 citations
TL;DR: In this article, it was shown that the semigroup of quotients Q = Q(S) corresponding to the filter of dense right ideals of a semigroup S is also a semilattice of groups.
TL;DR: In this paper, it was shown that the hypothesis of Theorem I does not imply that T is holomorphic on (0, oo) unless the inequality on the differences is assumed for all nh.
Abstract: Suppose T is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on [0, oo). It is known that lim supx_O I T(x) II 2 implies T is holomorphic on (0, oo). Theorem I is a generalization of this as follows: Suppose M > 0, 0 0, then there exists b > 0 such that T is holomorphic on [b, oo). Theorem II shows that, in some sense, b 0 as r 0. Theorem I is an application of Theorem III: Suppose M > 0, 0 0, [t, t + nh] C [4s, 4s], then f has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all nh. An example is given to show the hypothesis of Theorem I does not imply T holomorphic on (0, oo).
01 Jan 1974
TL;DR: In this paper, the authors identify many of the important subsets of the set of 2x2 matrices with complex number elements, having under matrix multiplication a semigroup structure.
Abstract: This is an identification of many of the important subsets of the set of 2x2 matrices with complex number elements, having under matrix multiplication a semigroup structure. The concepts defined are important and the structures noted are excellent examples for use in mathematics education. Some applications are noted in the references. Since we are concerned only with 2x2 matrices the results are easily verified and the proofs are omitted.