Showing papers on "Bicyclic semigroup published in 1986"

On the representations of the full matrix semigroup on homogeneous polynomials

Abstract: It is known that in a free inverse semigroup I the problem of equality of words is decidable [I]. It has been established in [2] that the elementary theory of I is decidable in the case when I is monogenic and inseparable in other cases. For finitely generated nonmonogenic free inverse semigroups the undecidability of positive theories was proved in [3]. The main goal of the present note is to prove that the problem of consistency of systems of equations on a nonmonogenic free inverse semigroup is undecidable.

Diophantine theories of free inverse semigroups

Abstract: It is known that in a free inverse semigroup I the problem of equality of words is decidable [I]. It has been established in [2] that the elementary theory of I is decidable in the case when I is monogenic and inseparable in other cases. For finitely generated nonmonogenic free inverse semigroups the undecidability of positive theories was proved in [3]. The main goal of the present note is to prove that the problem of consistency of systems of equations on a nonmonogenic free inverse semigroup is undecidable.

Inverse Semigroup Algebras

Abstract: A survey is given of work on inverse semigroup algebras concerning semiprimitivity, nonexistence of nonzero nil ideals, von Neumann regularity, and (for contracted algebras) simplicity.

On the radical of semigroup algebras satisfying polynomial identities

Abstract: In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebra K [ S ] of an arbitrary semigroup S over a field K in the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties of S or, more precisely, by some groups derived from S . For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

Semigroups with complemented congruence lattices

Abstract: Using the decomposition of a semigroup into itsI-classes, the paper gives a characterisation of all globally idempotent semigroups whose lattice of congruences is complemented. Furthermore, an arbitrary semigroup has a complemented congruence lattice if and only if it is an inflation of a semigroup characterised in this way. Thus the general problem of describing all semigroups with complemented congruence lattices is reduced to that of studying the question for the class of simple semigroups.

A Description of the Jacobsen Radical of Semigroup Rings of Commutative Semigroups

Abstract: The aim of this paper is to give a review of the results concerning the description of the Jacobson radical of semigroup rings of commutative semigroups. As an application we study when such semigroup rings are semilocal or local. In the final section we show that the results on the Jacobson radical are also valid for other concrete radicals.

Properties of relatively free inverse semigroups

Abstract: The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if S is combinatorial (i.e., X is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S does not contain a copy of the bicyclic semigroup) or E-unitary (i.e., E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup

Proper embeddability of inverse semigroups

Abstract: Let S be an inverse semigroup. We prove that there is a ring with a proper involution * in which S is *-embeddable. The ring will be a natural one, R[S], the semigroup ring of S over any formally complex ring R; for example ℝ, Ȼ.

The lattice of r-unipotent congruences on a regular semigroup

Abstract: If S is an inverse semigroup then E is a congruence on C(S). If S is a regular semigroup then each E -class of C(5) is a complete modular sublattice of C(5). (See [6]. ) In [5, Sec. 3] Petrich presents a few characterisations of E when S is an inverse semigroup. Here we prove that on a regular semigroup 5, the relation E restricted to RC(5) is a congruence. Also we extend Petrichfs results to the lattice RC(S) of a regular semigroup 5 and present a characterisation of the greatest element of each E-class. Characterisations of the least element of each E-class have been presented by Feigenbaum [I] and La Torte[4]. THE LATTICE RC(S) We use, whenever possible, the notation of Howie [3]. Recall first that a regular semigroup 5 is said to be R-u_~potent if its set of idempotents E(S) is a left reqular band, i.e. if E(5) satisfies the identity ere = el. In [7,1; 8, 1.1 ] it is shown that on a regular semigroup 5,

The Group of Units of A Commutative Semigroup Ring of A Torsion-Free Semigroup

Abstract: We seek necessary and sufficient conditions for the units of a commutative semigroup ring R[S] to be determined by the nilradical of R[S] and the units of R[G] where G is the group of units of S. We assume that R is a commutative ring with identity and S is a torsion-free semigroup with identity.