Semigroups and their natural order
H. Mitsch · Czech digital mathematics library · 1994
For any semigroup (S, •) the relation: a < b if an only if a = xb = by, xa = a for some x,y £ S 1 is called the natural partial order of S (see [8]).The relationship between the structure of S and certain properties of its natural partial order is investigated: trivial or total order, principal order ideals defined by ^-related elements, compatibility with multiplication, primitive or completely simple or group congruences, retract extensions of regular semigroups, and strong semilattices of semigroups.