Abstract An element e of an ordered semigroup (S, , ) S, ·, ≤ is called idempotent (resp. generalised idempotent) if e {e^2} e ≤ e 2 (resp. (e, {e^2}) { } e, e 2 ∈ ℜ ≤ where { } ℜ ≤ is the smallest congruence on S containing the relation { ^-1} ≤ ∪ ≤ - 1). The set of all idempotents (resp. generalised idempotents) of S is denoted by E (S) E S (resp. {E^G} (S) E G S). S is called orthodox if (i) the set E (S) E S is non empty and (ii) ef {E^G} (S) e f ∈ E G S for every e, f E (S) e, f ∈ E S. An element x in S is an inverse (resp. generalised inverse) of an element a of S if a axa a ≤ a x a and x xax
Michael Tsingelis (Mon,) studied this question.