On characters and characteristic polynomials of transformation semigroups (Q1078337)

Denote by \(h_ X: S\to C\) the transformation character of a finite semigroup S left acting on a set X (i.e. X is a left S-set), where C is the field of complex numbers. The author characterizes elements of a 0- simple semigroup S such that \(h_ X(s)=h_ X(t)\) for all left S-sets X and proves that for any pair s,t\(\in S\) we have \(h_ X(s)=h_ X(t)\) for every S-set X iff the characteristic polynomials \(f_{s,X}\), \(f_{t,X}\) of endomorphisms of the left module over the semigroup algebra CS of S over C freely generated by X, which are induced by s, t, respectively, coincide for any S-set X.