A semigroup S is called F-monoid if S has an identity and if there exists a group congruence rho on S such that each rho-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 "*", "o", and "-" on S defined by means of the greatest elements in the different rho-classes of S are studied. Using their properties a charaterization of F-monoids S by their regular part S degrees = {a degrees a is an element of S} and the associates of elements in S degrees is given. Under the hypothesis that S* = {a* a is an element of S} is a subsemigroup it is shown that S is regular, whence of a known structure ( see Giraldes et al., 2004).
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