Shift invariant algebras, Segre products and regular languages

Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of nonnegative integers by shifting variables. Furthermore, we exhibit a filtration of shift invariant monomial algebras that has a rational equivariant Hilbert series, but whose presentation ideals do not stabilize.