1,030 research outputs found
Relatively Congruence-free Regular Semigroups
Yu, Wang, Wu and Ye call a semigroup S τ -congruence-free, where τ is an equivalence relation on S, if any congruence ρ on S is either disjoint from τ or contains τ . A congruence-free semigroup is then just an ω-congruence-free semigroup, where ω is the universal relation. They determined the completely regular semigroups that are τ -congruence-free with respect to each of the Green’s relations. The goal of this paper is to extend their results to all regular semigroups. Such a semigroup is J –congruence-free if and only if it is either a semilattice or has a single nontrivial J -class, J, say, and either J is a subsemigroup, in which case it is congruence-free, or otherwise its principal factor is congruence-free. Given the current knowledge of congruence-free regular semigroups, this result is probably best possible. When specialized to completely semisimple semigroups, however, a complete answer is obtained, one that specializes to that of Yu et al. A similar outcome is obtained for L and R. In the case of H, only the completely semisimple case is fully resolved, again specializing to those of Yu et al
SEMIGRUP REGULER DAN SIFAT-SIFATNYA
This article discusses some properties of regular semigroups. These properties are especially concerned with the relation of the regular semigroups to ideals, subsemigroups, groups, idempoten semigroups and invers semigroups. In addition, this paper also discusses the Cartesian product of two regular semigroups.
Keywords:ideal, idempoten semigroup, inverse semigroup, regular semigroup, subsemigroup
F-regular semigroups
A semigroup S is called F-regular if S is regular and if there exists a group congruence rho on S such that every rho-class contains a greatest element with respect to the natural partial order of S (see [K.S. Nambooripad, Proc. Edinburgh Math. Soc. 23 (1980) 249-260]). These semigroups were investigated in [C.C. Edwards, Semigroup Forum 19 (1980) 331-345] where a description similar to the F-inverse case (see [R. McFadden, L. O'Carroll, Proc. London Math. Soc. 22 (1971) 652-666]) is given. Further characterizations of F-regular semigroups, including an axiomatic one, are provided. The main objective is to give a new representation of such semigroups by means of Szendrei triples (see [M. Szendrei, Acta Sci. Math. 51 (1987) 229-249]). The particular case of F-regular semigroups S satisfying the identity (xy)* = y*x*, where x* epsilon S denotes the greatest element of the rho-class containing x epsilon S, is considered. Also the F-inversive semigroups, for which this identity holds, are characterized. (C) 2004 Elsevier Inc. All rights reserved.Fundação para a Ciência e a Tecnologia (FCT) - POCTI
Finite complete rewriting systems for regular semigroups
It is proved that, given a (von Neumann) regular semigroup with finitely many
left and right ideals, if every maximal subgroup is presentable by a finite
complete rewriting system, then so is the semigroup. To achieve this, the
following two results are proved: the property of being defined by a finite
complete rewriting system is preserved when taking an ideal extension by a
semigroup defined by a finite complete rewriting system; a completely 0-simple
semigroup with finitely many left and right ideals admits a presentation by a
finite complete rewriting system provided all of its maximal subgroups do.Comment: 11 page
Varieties of \u3cem\u3eP\u3c/em\u3e-Restriction Semigroups
The restriction semigroups, in both their one-sided and two-sided versions, have arisen in various fashions, meriting study for their own sake. From one historical perspective, as “weakly E-ample” semigroups, the definition revolves around a “designated set” of commuting idempotents, better thought of as projections. This class includes the inverse semigroups in a natural fashion. In a recent paper, the author introduced P-restriction semigroups in order to broaden the notion of “projection” (thereby encompassing the regular *-semigroups). That study is continued here from the varietal perspective introduced for restriction semigroups by V. Gould. The relationship between varieties of regular *-semigroups and varieties of P-restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox *-semigroups and varieties of “orthodox” P-restriction semigroups, leading to concrete descriptions of the free orthodox P-restriction semigroups and related structures. Specializing further, new, elementary paths are found for descriptions of the free restriction semigroups, in both the two-sided and one-sided cases
Cross-connections of the singular transformation semigroup
Cross-connection is a construction of regular semigroups using certain
categories called normal categories which are abstractions of the partially
ordered sets of principal left (right) ideals of a semigroup. We describe the
cross-connections in the semigroup of all non-invertible
transformations on a set . The categories involved are characterized as the
powerset category and the category of partitions . We
describe these categories and show how a permutation on gives rise to a
cross-connection. Further we prove that every cross-connection between them is
induced by a permutation and construct the regular semigroups that arise from
the cross-connections. We show that each of the cross-connection semigroups
arising this way is isomorphic to . We also describe the right
reductive subsemigroups of with the category of principal left ideals
isomorphic to . This study sheds light into the more general
theory of cross-connections and also provides an alternate way of studying the
structure of
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