Shirota's and Milgram's (in fact, also Kaplansky's) results characterizing compact or realcompact spaces by means of semigroups \(C(X)\), are generalized to semigroups \(C(X,S)\) for special semigroups \(S\) (the reviewer's generalization of the above mentioned results [Math. Z. 111, 214--220 (1969; Zbl 0175.49602)], is not covered). We quote the following result, selected because of its simpler formulation: If \(X, Y\) are Tikhonov spaces, \(S_1,S_2) are homeomorphic provided \(C(X,S_1)\), \(C(X,S_2)\) are isomorphic as semigroups.