Character Theory of Monoids over an Arbitrary Field

The basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of Munn, Ponizovsky, McAlister, Rhodes and Zalcstein. In particular, McAlister determined the space of functions spanned by the irreducible characters of a finite monoid over $\mathbb{C}$ and the ring of virtual characters. In this paper, we present the corresponding results over an arbitrary field. As a consequence, we obtain a quick proof of the theorem of Berstel and Reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid. This is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics.