Orthodox ordered semigroups

An element e of an ordered semigroup ( S,· ,≤ ) S , · , ≤ is called idempotent (resp. generalised idempotent) if e≤ e² e ≤ e 2 (resp. ( e,e² ) ∈ ≤ 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. EG( S ) E G S ). S is called orthodox if (i) the set E( S ) E S is non empty and (ii) ef∈ EG( 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