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 $$x\le xax$$ x ≤ x a x (resp. $$\left( a,axa \right) ,\left( x,xax \right) \in {{\Re }_{\le }}$$ a , a x a , x , x a x ∈ ℜ ≤ ). We study the notions of generalised inverse and generalised idempotent element and we show that, in an orthodox ordered semigroup, if we know a single generalised inverse of an element a , then we know the set of all generalised inverses of a . We also study the structure of orthodox ordered semigroups giving basic properties of orthodox ordered semigroups and equivalent conditions (based on inverse, generalised inverse, idempotent, generalised idempotent elements) according to which an ordered semigroup is orthodox.