  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.
