Exponential families with semigroup-valued sufficient statistics

We consider an algebraic extension of the notion of an exponential family where the sufficient statistic takes values in an Abelian semigroup rather than a vector space. The characters of the semigroup play a role of exponential functions. As opposed to standard exponential families, characters are not necessarily everywhere positive. Still we show how basic results concerning existence and uniqueness of the MLE can be established, in some sense yielding a simpler theory. We shall describe examples which are very different from standard families and others which appear as natural extensions of standard families. In particular we shall examine graphical and hierarchical log-linear models in this light.