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MATHEMATICA BOHEMICA, Vol. 130, No. 2, pp. 203-220 (2005)
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Generalized $F$-semigroups

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E. Giraldes, P. Marques-Smith, H. Mitsch

* Emilia Giraldes*, UTAD, Dpto. de Matematica, Quinta de Prados, 5000 Vila Real, Portugal; * Paula Marques-Smith*, Universidade do Minho, Centro de Matematica, Campus de Gualtar, 4700 Braga, Portugal; * Heinz Mitsch*, Universit\"{a}t Wien, Institut f\"{u}r Mathematik, Nordbergstrasse 15, 1090 Wien, Austria, e-mail: ` heinz.mitsch@univie.ac.at`

**Abstract:** A semigroup $S$ is called a * generalized* $F$-* semigroup* if there exists a group congruence on $S$ such that the identity class contains a greatest element with respect to the natural partial order $\leq_{S}$ of $S$. Using the concept of an * anticone*, all partially ordered groups which are epimorphic images of a semigroup $(S,\cdot,\leq_{S})$ are determined. It is shown that a semigroup $S$ is a generalized $F$-semigroup if and only if $S$ contains an anticone, which is a principal order ideal of $(S,\leq_{S})$. Also a characterization by means of the structure of the set of idempotents or by the existence of a particular element in $S$ is given. The generalized $F$-semigroups in the following classes are described: monoids, semigroups with zero, trivially ordered semigroups, regular semigroups, bands, inverse semigroups, Clifford semigroups, inflations of semigroups, and strong semilattices of monoids.

**Keywords:** semigroup, natural partial order, group congruence, anticone, pivot

**Classification (MSC2000):** 20M10

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