Les F-reguliers a gauche

1. The concept of left F-regular semigroups was first defined by Batbedat at the Oberwolfach meeting in 1981. It generalizes the notion of F-regular semigroup introduced by Edwards 4], itself a generalization of the F-inverse semigroups defined by McFadden/O’Carroll 6]
2. In the present paper we generalize the results of 4] and 6] by defining two preorders and ? on a monoid S with a distinguished band E, as follows: iff x=ay for some a∈E xδy iff x=yb for some b∈E
3. When S is regular orthodox and E=E(S), is the preorder of 1] p. 29 and is the order of 1] p. 31 (the order of 4]): in fact is the natural partial order introduced by Nambooripad 7].
4. In b), we define the relation Σ on S: xΣy iff exe=eye for some e∈E Then we consider the congruence σ generated by Σ.
5. DEFINITION. S is left F_E-monoid if each σ-class contain a greatest element with respect to .
6. PARTICULAR CASES. When S is regular, S is left F_E-regular. When S is regular orthodox and E=E(S), S is left F-regular.
7. We describe the structure of left F-regular semigroups like in 1], 2], 4] and 6]. Note that every left F-regular semigroup is a gammasemigroup 3]
8. Particular Cases (gamma morphism) and applications (congruences).