Shift invariant algebras, Segre products and regular languages

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