A natural partial order for semigroups

Abstract

A partial order on a semigroup ( S , ⋅ ) (S, \cdot ) is called natural if it is defined by means of the multiplication of S S . It is shown that for any semigroup ( S , ⋅ ) (S, \cdot ) the relation a ≤ b a \leq b iff a = x b = b y a = xb = by , x a = a xa = a for some x x , y ∈ S 1 y \in {S^1} , is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup ( T X , ∘ ) ({T_X}, \circ ) of all maps on the set X X are investigated.