Tannakian categories with semigroup actions

Ostrowski’s theorem implies that $\log (x),\log (x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $$\mathbb{N}$^n\times $\mathbb{Z}$/n_1$\mathbb{Z}$\times\ldots\times$\mathbb{Z}$/n_r$\mathbb{Z}$$ on Tannakian categories. This is the class of semigroups that appear in many applications.