Semigroups whose endomorphisms are power functions. (Q2447223)

It is proved that a finite commutative semigroup \(S\) is a cyclic semigroup if and only if every endomorphism \(f\) of \(S\) is equal to a power function, that is, \(f(x)=x^n\) for some positive integer \(n\). This result cannot be extended to infinite commutative semigroups. The second result concerns finite commutative semigroups \(S\) with \(1\neq 0\): ``every endomorphism of \(S\) preserving \(0\) and \(1\) 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''.