Showing papers on "Bicyclic semigroup published in 1971"
[...]
45 citations
28 citations
TL;DR: A finitely generated semigroup S given by generators and the set of relations between these generators is called an F-semigroup if it is isomorphic to a subsemigroup of a free semigroup as discussed by the authors.
Abstract: A finitely generated semigroup S given by generators and the set of relations between these generators is called an F-semigroup if it is isomorphic to a subsemigroup of a free semigroup. In this paper three theorems concerning F-semigroups are proven and some unsolved problems are presented.
20 citations
TL;DR: In this paper, it was shown that if B is the bicyclic semigroup, then a certain inverse subsemigroup F of B × B is a free elementary inverse semigroup.
Abstract: The purpose of this note is to give a characterization of a free inverse semigroup on a singleton set. It will be shown that if B is the bicyclic semigroup, then a certain inverse subsemigroup F of B × B is a free elementary inverse semigroup.
19 citations
18 citations
TL;DR: The non-commutative version of this theorem is unsettled as discussed by the authors, and it is not known whether a group with zero is a R-semigroup unless it admits a ring structure.
Abstract: A multiplicative semigroup S with 0 is said to be a R-semigroup if S admits a ring structure. Isbell proved that if a finitely generated commutative semigroup is a R-semigroup, then it should be finite. The non-commutative version of this theorem is unsettled. This paper considers semigroups, not necessarily commutative, which are principally generated as a right ideal by single elements and semigroups which are generated by two independent generators and describes their structure. We also prove that if a cancellative 0-simple semigroup containing an identity is a R-semigroup, then it should be a group with zero.
13 citations
12 citations
TL;DR: In this paper, it was shown that all finite dimensional representations of a completely 0-simple semigroup S over a field Φ can be obtained as extensions of those of its maximal subgroups and has given a method for constructing all such representations.
Abstract: A. H. Clifford [1], [2] has shown that all finite dimensional representations of a completely 0-simple semigroup S over a field Φ.phi; can be obtained as extensions of those of its maximal subgroups and has given a method for constructing all such representations. This representation theory depends strongly on the fact the representations under consideration are finite dimensional and is not adequate to deal with the infinite dimensional case or with representations over arbitrary rings. In order to determine the structure of the (contracted) algebra Φ( S ) of S modulo its radical, one has to consider representations which are not finite dimensional or over fields; c.f. ‘6’. Hence Clifford's theory does not suffice for this purpose.
10 citations
TL;DR: In this paper, the equivalence classes induced by the Green relations on the elements of a finite semigroup are computed for the first time, and algorithms are described for computing these classes.
Abstract: In this paper algorithms are described for computing the equivalence classes induced by the Green relations on the elements of a finite semigroup.
9 citations
TL;DR: In this paper, the maximal right quotient semigroup of a semilattice of groups is studied and a structure theorem for such semigroups is given by Theorem 4.11 of [2].
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].
7 citations
TL;DR: In this paper, the authors considered the problem of finding the maximal subgroups of a compact abelian group with identity and showed that the maximal groups of a commutative version of Ext (B, G) are always a union of groups.
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.
TL;DR: In this article, the authors give a structure for a semigroup which is a semilattice of bisimple inverse semigroups, and characterize the idempotent separating congruences.
Abstract: The purpose of this paper is to give a structure for a semigroup which is a semilattice of bisimple inverse semigroups and satisfies certain conditions. For such a semigroup, we characterize the idempotent separating congruences.
TL;DR: In this paper, it was shown that a quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if EX is simple.
Abstract: In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.
TL;DR: The following problem was posed by the second author in 'Semigroup Forum' as discussed by the authors, Vol. 1, No. 1, No. 2, 1970, p. 91:
Abstract: The following problem was posed by the second author in ‘Semigroup Forum’, Vol. 1, No. 1, 1970, p. 91:
TL;DR: In this article, it was shown that for any set A of metric semigroups there exists a metric semigroup U such that each S in A is topologically isomorphic to a subsemigroup of U.
Abstract: J.H. Michael recently proved that there exists a metric semigroup U such that every compact metric semigroup with property P is topologically isomorphic to a subsemigroup of U ; where a semigroup S has property P if and only if for each x, y in S , x ≠ y , there is a z in S such that xs ≠ yz or zx ≠ zy A stronger result is proved here more simply. It is shown that for any set A of metric semigroups there exists a metric semigroup U such that each S in A is topologically isomorphic to a subsemigroup of U . In particular this is the case when A is the class of all separable metric semigroups, or more particularly the class of all compact metric semigroups.
TL;DR: The equivalence relation 3 on the state space, analogous to the .~-relation of Green, has proved to be of considerable importance in studying actions.
Abstract: Given a topological semigroup T and a space X, an act is a continuous function of T x X into X (denoted by juxtaposition) such that (s t )x = s( tx) for each s, t e T and each x e X. We say that T acts on X and refer to X as the state space of the act. T acts unitarily on X if and only if x ~ Tx for each x e X. The equivalence relation 3 on the state space, analogous to the .~-relation of Green, has proved to be of considerable importance in studying actions. It is defined by
01 Jan 1971
TL;DR: In this article, the authors give a complete description of all compact totally 3C ordered semigroups and a special case of the structure theorem for Psemigroups, which is called abelian.
Abstract: Compact totally 3C ordered semigroups are charac- terized. Each such semigroup is abelian and is, in fact, a closed subsemigroup of an /-semigroup. Several questions are posed about (algebraic) semigroups which are naturally totally (quasi-) ordered. The structure of Psemigroups (or standard threads) has been known for some time (3), (5), (lO). One of the first steps in deter- mining this structure was made by Faucett in showing that any I- semigroup is totally 3C ordered (5, Lemma 2). In this paper we give a complete description of all compact totally 5C ordered semigroups. A consequence of the structure theorem is that each such semigroup is a closed subsemigroup of an Psemigroup and is hence abelian. The structure theorem for Psemigroups is a special case of our theorem. Several questions are posed at the end of the paper. An Psemigroup is a topological semigroup on an arc in which one endpoint acts as an identity and the other acts as a zero. If 5 is a semigroup then 51 is S if S has an identity and 51 is 5 with an identity adjoined otherwise. Following (7), we define the following