Showing papers on "Cancellative semigroup published in 1970"

Regularity of semigroup rings

Abstract: The conditions unlder which a semnigroup ring is regu- lar (in the sense of von Neumann) are inlvestigated. SLufficient con- ditions are obtained in order that the semnigroup ring of an inverse semigroup be regular. Consequences of the regularity of the seni- group ring for the subgroups of the semigroup are established. The two results are then used to find necessary and sufficien-t conditions for the regularity of the semigroup ring when the semigroup is inverse and a union of groups. 1. Introduction. Regularity of group rings has been investigated by Auslander (1), Connell (31 and McLaughlin (4). In this paper we attempt to extend their results to semigroup rings. Throughout this paper R will denote an associative ring having an identity element. If R is a ring and D is a semigroup RD will denote the semigroup ring of D over R and (RD)o the contracted semiigroup ring if D has a zero. The definition of (contracted) seniigroup rings is given in (8, ?3). It is analogous to the definition of semnigroup algebra

Embedding topological semigroups in topological groups

Abstract: If we consider a semigroup, its algebraic structure may be such that it is isomorphic to a subsemigroup of a group, or is algebraically embeddable in a group. This problem was investigated in 1931 by Ore who obtained in (4) a set of necessary conditions for this embedding. A necessary condition is that the semigroup should be cancellative: for any a, x, y in the semigroup either xa = ya or ax = ay implies that x = y. Malcev in (3) showed that this was not sufficient. It is enough to note that his example was a non-commutative semigroup: a commutative cancellative semigroup is embeddable algebraically in a group.

The Maximal Semilattice Decomposition of a Semigroup, Radicals and Nilpotency

Abstract: In paper [7] M. Petrich dealt with the maximal semilattice decomposition of a semigroup and he studied the classes of this decomposition. In the present paper a description of these classes and their products is given in terms of Luh Jiang completely prime radicals and faces of a semigroup S. Also the case of the commutative semigroup is discussed. The last, 5th section is self-containe d. Here a characterizati on of the class of all periodic semigroups with period 1, a characterization of the class of all periodic semigroups with index 1 and some characterizations of the class of all bands are given. We accomplished this using the mappings M-> Nt(M) (i = = 1,2, 3), where N±(M) (N2(M)) [N3(M)] is the set of all strongly (weakly) [almost] nilpotent elements with respect to the subset M of the semigroup S (see [9]).

Congruences on regular semigroups

Abstract: The kernel of a congruence on a regular semigroup S may be characterized as a set of subsets of S which satisfy the Teissier-Vagner-Preston conditions. A simple construction of the unique congruence associated with such a set is obtained. A more useful characterization of the kernel of a congruence on an orthodox semigroup (a regular semigroup whose idempotents form a subsemigroup) is provided, and the minimal group congruence on an orthodox semigroup is determined.

Cancellative commutative semigroups

Abstract: The study of t.c.r. (=totally cancellative reduced) real semigroups (which are just convex cones in real vector spaces, with the induced addition) enables us to answer some questions about t.c.r. semigroups in general. For example, a finite-dimensional divisible commutative semigroups is locally free if and only if it is t.c.r. and has the Riesz Interpolation Property.

Representations of certain compact semigroups by -semigroups

Abstract: An HL-semigroup is defined to be a topological semigroup with the property that the Schutzenberger group of each XI-class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL-semigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HL-semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL-semigroups to separate points of S. Introduction. A semigroup analog of the theorem of Peter and Weyl is not in existence at the present time. Indeed, such a theory for compact semigroups closely paralleling that for compact groups is generally believed to be unfeasible. In this paper an alternate approach is considered. The alternative is to replace groups of nonsingular complex matrices by compact semigroups with the property that the Schiutzenberger group of each X-class is a Lie group. Such semigroups are called HL-semigroups. The following question is considered: Given a compact semigroup S, do there exist enough homomorphisms of S into HL-semigroups to separate the points of S? Furthermore, can these homomorphisms be chosen so as to preserve the XV'-class structure of S? We follow the current trend and call a homomorphism from S into an HL-semigroup T a representation of S by the HLsemigroup T. The main result of this paper is that every irreducible semigroup admits enough X#'-class separating representations by HL-semigroups to separate points in S. Moreover, if S/I is separable, then each of the HL-semigroups may be chosen to have finite dimension. ?1 is devoted to preliminary results of a general nature. ?2 deals with irreducible semigroups. For the most part we will use the terminology and notation of [7]. All semigroups, homomorphisms, and isomorphisms will be in the category of compact semigroups and continuous homomorphisms. The authors are indebted to J. D. Lawson, Michael W. Mislove, and Eleanor Bailey for their useful comments and suggestions. Received by the editors January 30, 1969 and, in revised form, August 11, 1969. AMS Subject Classifications. Primary 2205; Secondary 2092, 2210, 2250.

On input semigroups of automata

Abstract: In this paper, we consider the problem of what topological semigroups can serve as input semigroups of what (topological) automata. A semigroup is said to be admissible if it serves as an input semigroup of a non-trivial “strongly connected” automaton that has a distinguishable state (see Definition 2). For the discrete or the compact case, the class of all the admissible semigroups is fully characterized: a discrete or compact topological semigroup (I, m) is admissible if and only if there exists a closed congruence relationR such that the quotient semigroup (I/R, m R ) is non-trivial, right simple, and left unital. This work stems from Weeg's [10], who considered a similar problem in the discrete case.

Positive Clifford semigroups on the plane

Abstract: This work is devoted to a preliminary investigation of positive Clifford semigroups on the plane. A positive semigroup is a semigroup which has a copy of the nonnegative real numbers embedded as a closed subset in such a way that 0 is a zero and 1 is an identity. A positive Clifford semigroup is a positive semigroup which is the union of groups. In this work it is shown that if S is a positive Clifford semigroup on the plane, then each group in S is commutative. Also, a necessary and sufficient condition is given in order that S be commutative, and an example is given of such a semigroup which is, in fact, not commutative. In addition, both the number and the structure of the components of groups in S is determined. Finally, it is shown that S is the continuous isomorphic image of a semilattice of groups. A topological semigroup is a Hausdorff space together with a continuous associative multiplication. A real semigroup has been defined by J. G. Horne, Jr. [4] to be a topological semigroup containing a subsemigroup R iseomorphic to multiplicative semigroup of real numbers, embedded as a closed subset of E2 in such a way that 1 is an identity and 0 is a zero. Similarly, the author has defined a positive semigroup to be a topological semigroup containing a subsemigroup N iseomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E2 so that 1 is an identity and 0 is a zero [2]. Relying heavily on the work done by Horne in [4] and [5], this work is devoted to a study of positive semigroups on E2 with the additional requirement that these semigroups be the union of groups. Let us call such semigroups positive Clifford semigroups [3]. We will show that if S is a positive Clifford semigroup on E2, then each group in S is commutative. Also, we will give a necessary and sufficient condition in order that a positive Clifford semigroup on E2 be commutative, and we will give an example of a positive Clifford semigroup on E2 which is, in fact, not commutative. We will show that each group in a positive Clifford semigroup S on E2 has one, two, or four components, that each two dimensional group is P x P, P x P x {1,, or P x P x F, where F is the four group, and that each one dimensional group is P, P x {1, 1}, or P x F. Also, we will characterize S in terms of the sector of identity Presented to the Society, November 8, 1968; received by the editors May 19, 1969 and, in revised form, January 8, 1970. AMS 1969 Subject Classifications. Primary 2205.