Les F-reguliers a gauche

Abstract

• a)- 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]

• b)- 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

• c)- 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].

• d)- 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.

• e)- DEFINITION. S is left F_E-monoid if each σ-class contain a greatest element with respect to

  

  .

• f)- 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.

• g)- 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]

• h)- Particular Cases (gamma morphism) and applications (congruences).