51 research outputs found
F −semigroups
A semigroup S is called F−semigroup if there exists a group congruence ρ on S such that every ρ −class contains a greatest element with respect to the natural partial order ≤ of S . This generalizes the concept of F−inverse semigroups introduced by V. Wagner
in 1961 and investigated by McFadden and O’Caroll in 1971. Five different characterisations of general F−semigroups S are given: by means of residuals, by special principal anticones, by properties of the set of idempotents, by the maximal elements in (S, ≤) and finally, an axiomatic one using an additional unary operation. Also, F−semigroups in special classes are considered; in particular, inflations of semigroups and strong semi-
lattices of monoids are studied.Centro de Matemática da Universidade do MinhoFundação para a Ciência e a Tecnologia (FCT
F-monoids
A semigroup is called if has an identity and if there exists a group congruence on such that each -class of contains a greatest element with respect to the natural partial order of (see Mitsch, 1986). Generalizing results given in Giraldes et al. (2004) and specializing some of Giraldes et al. (Submitted) five characterizations of such monoids are provided. Three unary operations , and on defined by means of the greatest elements in the different -classes of are studied. Using their properties, a charaterization of -monoids by their regular part and the associates of elements in is given. Under the hypothesis
that is a subsemigroup it is shown that is regular, whence of a
known structure (see Giraldes et al., 2004).Fundação para a Ciência e a Tecnologia (FCT
Associate inverse subsemigroups of regular semigroups
By an associate inverse subsemigroup of a regular semigroup S we mean
a subsemigroup T of S containing a least associate of each x ∈ S, in relation to
the natural partial order ≤S. We describe the structure of a regular semigroup with
an associate inverse subsemigroup, satisfying two natural conditions. As a articular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element.Fundação para a Ciência e a Tecnologia (FCT
F–semigroups
A semigroup S is called F- semigroup if there exists a group-congruence ρ on S such that every ρ-class contains a greatest element with respect to the natural partial order ≤S of S (see [8]). This generalizes the concept of F-inverse semigroups introduced by V. Wagner [12] and investigated in [7]. Five different characterizations of general F-semigroups S are given: by means of residuals, by special principal anticones, by properties of the set of idempotents, by the maximal elements in (S,≤S) and finally, an axiomatic one using an additional unary operation. Also F-semigroups in special classes are considered; in particular, inflations of semigroups and strong semilattices of monoids are studied
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
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