Abstract
An abundant semigroup is called F-abundant if each class modulo the minimum cancellative congruence has a maximum element with respect to the natural order. This paper studies a kind of F-abundant semigroup in which all these maximum elements form a sub-semigroup and the set of idempotents is a semilattice. After the characterizations of such semigroup are analysed, the structure theorem is obtained by the semidirect product of a semilattice and a cancellative monoid.
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