Showing papers on "Cancellative semigroup published in 1982"

The structure of pseudo-inverse semigroups

Abstract: A regular semigroup S is called a pseudo-inverse semigroup if eSe is an inverse semigroup for each e= e2 C S. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory. Completely (0-) simple semigroups and inverse semigroups form the first prototypes for the study of pseudo-inverse semigroups. We therefore can say that the theory of regular semigroups began with the study of pseudo-inverse semigroups [40, 45]. We may distinguish four successful trends in the papers which since then have dealt with some wider classes of pseudo-inverse semigroups: 1. the subdirect products of completely 0-simple and completely simple semigroups, 2. the generalized inverse semigroups (orthodox pseudo-inverse semigroups, 3. the normal band compositions of inverse semigroups, and 4. Rees matrix semigroups over inverse semigroups (with zero). Subdirect products of completely 0-simple semigroups and completely simple semigroups were initiated in [13, Chapter 2] and studied in great detail in [18] (see also ?4 of [14]); this class contains several interesting subclasses: (a) the trees of completely 0-simple semigroups [18] which include the primitive regular semigroups [7, Vol. II, 16, 39, 44, 46], (b) the regular locally testable semigroups [50] which include the normal bands [36] and the combinatorial completely 0-simple semigroups, (c) the normal bands of groups [37] which include the semilattices of groups [7, Vol. I], (d) the subdirect products of Brandt semigroups which include the locally testable semigroups which are inverse semigroups [50] and the primitive inverse semigroups [39]. The generalized inverse semigroups were introduced in [48] as a special class of orthodox semigroups; they include (a) the inverse semigroups, (b) the orthodox completely 0-simple semigroups [9] and the rectangular groups, (c) the Received by the editors October 25, 1979 and, in revised form, March 25, 1981 and June 18, 1981. AMS (MOS) subject classifications (1970). Primary 20M10.

The left regular *-representation of an inverse semigroup

Abstract: The left regular *-representation of the semigroup algebra of an inverse semigroup is faithful. Clifford semigroups with a particular type of semilattice are shown to have the weak containment property if and only if each subgroup is amenable. An inverse semigroup is a semigroup in which for each element s there exists a unique element, which we denote s*, such that ss*s = s and s*ss* =s*. From this it follows that the idempotents commute, products of idempotents are idempotents, and (st)* = t*s* [5, pp. 130-131]. For an inverse semigroup S, Es will denote the set of idempotents of S. ES is a lower semilattice under the operation e A f = ef . A Clifford semigroup is an inverse semigroup T whose idempotents are central; then T = U{Ge: e E ET} where Ge is the greatest subgroup of T containing e. We identify s E S with the function in 11(S) which is 1 at s and 0 elsewhere. Then l1(S) is a Banach algebra with multiplication the continuous bilinear extension of the semigroup multiplication, called the (11-)semigroup algebra of S. The involution * on S extends to a unique continuous involution * on 11(S) by conjugate linearity. Then l(S) is a Banach star algebra. Barnes [1] constructed the left regular *-representation of l1(S) (on 12(S)), which we will denote by Ls, by Ls(a)b = { ab if a*ab = b, i.e. if a*a > bb*,

Seminearrings, seminearfields and their semigroup-theoretical background

Abstract: In this paper we deal with some basic properties of seminearrings (S,+,·), in particular with respect to multiplicative cancellativity (§2) and to seminearfields (§3, §4). All seminearrings of order 2 are listed in §5. In general, we do not assume that (S,+) is associative. Moreover, we present most of our results as consequences of corresponding statements on semigroups (S,·). Clearly, all results apply to semirings and nearrings, particularly to those with associative addition, and some of them will be new also in this context.

P-systems in regular semigroups

Abstract: In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band ES of idempotents of S such that F∋e, ES∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of FS.

Characters of finite inverse semigroups

Abstract: For elements of a finite inverse semigroup, an equivalence relation called p-conugacy is introduced. It is proved that for any matrix representation of a finite inverse semigroup the values of the character of the representation are equal on p-conjugate elements. The number of inequivalent irreducible matrix representations of a finite inverse semigroup over the field of complex numbers is equal to the number of classes of p-conjugate elements.