Geodesic
$F$--semigroups
Algebra and discrete mathematics, no. 3 (2007), pp. 67-86.

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A semigroup $S$ is called $F$- semigroup 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 $\leq_S$ 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 $(S,\leq_S)$ 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.
Keywords: natural partial order, residual
Mots-clés : maximal elements, group congruence, anticone. 
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     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2007_3_a8/}
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Emilia Giraldes; Paula Marques-Smith; Heinz Mitsch. $F$--semigroups. Algebra and discrete mathematics, no. 3 (2007), pp. 67-86. http://geodesic.mathdoc.fr/item/ADM_2007_3_a8/