An analogue of the Kronecker — Cappelli theorem for systems of non-commutative linear equations generating linear languages

The paper continues the study of systems of noncommutative polynomial equations, which are interpreted as grammars of formal languages. Such systems are solved in the form of formal power series (FPS), which express non-terminal symbols in terms of the terminal symbols of the alphabet and are considered as formal languages. Each FPS is associated with its commutative image, which is obtained under the assumption that all symbols denote commutative variables, real or complex. In this paper, we consider equations that are linear in nonterminal symbols with polynomial coefficients in terminal symbols, which means that these systems generate linear formal languages. As is known, the compatibility of a system of noncommutative polynomial equations is not directly related to the compatibility of its commutative image, and therefore, as an analogue of the Kronecker — Cappelli theorem, it is only possible to obtain a sufficient condition for the inconsistency of a noncommutative system.