Orthodox ordered semigroups

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