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A natural partial order for semigroups

A partial order on a semigroup (S, ) is called natural if it is defined by means of the multiplication of S. It is shown that for any semigroup (S, ) the relation a < b iff a = xb = by, xa = a for some x,y E S1, 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 (Tx, o) of all maps on the set X are investigated.