Cancellative semigroup

About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.

The quotient semigroup of a semigroup that is a semilattice of groups

Abstract: Let Q(S) denote the maximal right quotient semigroup of the semigroup S as defined in [4]. In this paper, we initiate a study of Q(S) when S is a semilattice of groups. A structure theorem for such semigroups is given by Theorem 4.11 of [2].

The closedness of the generator of a semigroup

Abstract: We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace trans- forms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector-valued functions with respect to scalar measures.

Structure of the semigroup of semigroup extensions

Abstract: Let B denote a compact semigroup with identity and G a compact abelian group. Let Ext (B, G) denote the semigroup of extensions of G by B. We show that Ext (B, G) is always a union of groups. We show that it is a semilattice whenever B is. In case B is also an abelian inverse semigroup with its subspace of idempotent elements totally disconnected, we obtain a determination of the maximal groups of a commutative version of Ext (B, G) in terms of the extension functor of discrete abelian groups. This paper is a continuation of our paper [3] in which we consider the notion of an extension of a compact abelian group G by a compact semigroup B with identity. In that paper we show that the collection (of equivalence classes) of extensions of G by B is a commutative semigroup with identity under the usual \"Baer sum\" of extensions. As in [3] let us denote this semigroup of extensions by Ext (B, G). In our previous paper we obtained, among other results, some theorems relating to the problem as to when a particular extension in Ext (B, G) is an idempotent. We also characterized those extensions which belong to the maximal subgroup of Ext (B, G) containing some particular idempotent. In this paper we are interested in the same kind of problem from a more global point of view, e.g., we are interested in the semigroup structure of Ext (B, G). In particular we show that Ext (B, G) is always a union of groups. We show that Ext (B, G) is a semilattice whenever B is a semilattice but that it may or may not be isomorphic to B. In case B is a compact abelian inverse semigroup with its subspace of idempotents totally disconnected we are able to obtain a fairly complete description of the maximal subgroups of a commutative version of Ext (B, G) in terms of the groups Ext (77, K) where K~ is a maximal subgroup of B and 77~ is a quotient group of G (here the \"hat\" denotes the Prontrjagin dual). Since 77 and K are necessarily discrete this computes the structure of Ext in terms of the usual extension functor of discrete abelian group theory. For pertinent comments regarding the historical development of Ext see [3] or [6]. 1. Preliminaries. In this section we give the basic definitions and state explicit results from [3] which will be needed in this paper. Let B denote a compact (Hausdorff) semigroup with identity 1B and let G denote a compact abelian group with Received by the editors September 29, 1970. AMS 1968 subject classifications. Primary 2205; Secondary 2092.

Global determinism of clifford semigroups

Abstract: In this paper we shall give characterizations of the closed subsemigroups of a Clifford semigroup. Also, we shall show that the class of all Clifford semigroups satisfies the strong isomorphism property and so is globally determined. Thus the results obtained by Kobayashi [‘Semilattices are globally determined’, Semigroup Forum29 (1984), 217–222] and by Gould and Iskra [‘Globally determined classes of semigroups’ Semigroup Forum28 (1984), 1–11] are generalized.