bibkey: "fliess1974products"
authors: "Michel Fliess"
year: 1974
title: "Sur divers produits de séries formelles"
doi: "10.24033/bsmf.1777"
url: "https://www.numdam.org/item/10.24033/bsmf.1777.pdf"
claim: "Rational noncommutative formal series are closed under Hadamard product, with a tensor-product representation."
strata_touched: []
license: "citation-only"
triage: "anchor"

Sur divers produits de séries formelles

Bulletin de la Société Mathématique de France 102, 181–191. Section 2 assumes a finite nonempty alphabet and a commutative unital coefficient ring. Section 3, Proposition 1, printed page 183, states in particular that the Hadamard product of two rational series is rational; the author attributes this closure result to Schützenberger. The remark on printed page 187 explicitly gives the tensor-product representation for the Hadamard product.

For real linear word statistics filtered by a deterministic finite automaton, the coefficients are multiplied by the accepted-language indicator. This is the classical Hadamard construction. The block lift, chronological effect-word convention and finite-horizon parameter correspondence are explained in the existing FiniteWordReachability Blueprint narrative. No novelty or native permission realization follows from the cited closure theorem.