We present a characterization of the maximal compatible extensions of a given compatible partial order ≤ r on a unary algebra (A,f ). These extensions can be constructed by using the compatible linear extensions of ≤ r*, where (A*,f*) is the so called contracted quotient algebra of (A,f) and the compatible partial order ≤ r* on (A*,f*) is naturally induced by ≤ r. Using this characterization, we determine the intersection of the maximal compatible extensions of ≤ r.