A semigroup S is called F−monoid if S has an identity and if there exists a group congruence ρ on S such that each ρ-class of S contains a greatest element with respect to the natural partial order of S (see Mitsch, 1986). Generalizing results given in Giraldes et al. (2004) and specializing some of Giraldes et al. (Submitted) five characterizations of such monoids S are provided. Three unary operations ⋆, ∘ and − on S defined by means of the greatest elements in the different ρ-classes of S are studied. Using their properties, a charaterization of F-monoids S by their regular part S∘={a∘:a∈S} and the associates of elements in S∘ is given. Under the hypothesis
that S⋆={a⋆:a∈S} is a subsemigroup it is shown that S is regular, whence of a
known structure (see Giraldes et al., 2004).Fundação para a Ciência e a Tecnologia (FCT