AbstractAn operand X of a monoid S is called saturated if every generalized orbit in X is contained in a union of others. Every operand has a natural decomposition as a union of an operand admitting an irredundant cover by maximal generalized orbits and of a saturated operand. There is a descending chain of suboperands of an operand which leads to the definition of the saturation length of an operand. S has no saturated operands if and only if S satisfies the ascending chain condition on orbits.