Semigroups whose endomorphisms are power functions

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=x ^m is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomorphism of S preserving 1 and 0 is equal to a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. Immediate consequences of the results are, on the one hand, a characterization of commutative rings whose multiplicative endomorphisms are power functions given by Greg Oman in the paper (Semigroup Forum, 86 (2013), 272–278), and on the other hand, a partial solution of Problem 1 posed by Oman in the same paper.