Toric ideals in polynomial rings in infinitely many variables: stabilization and other properties

Pairs [toric ideal<->monomial algebra] associated to monomial maps between polynomial rings in infinitely many variables will be considered. We will introduce the concept of finiteness up to a shift operator on the indices of variables. In the Noetherian setting these two objects behave very similarly. Surprisingly, they do not necessary behave the same in the infinite world. For instance, it will be shown that there are monomial algebras that are finitely generated up to our shift operator, and their respective ideals require generators of any degree. The second part of the talk will be on computing the rational form of an Equivariant Hilbert series for these objects using regular languages and finite automata. If time permits, inspired by Segre and tensor products we will end on definitions of Segre tensor languages.