On finitely based systems of generalized polynomials

The theory of so-called generalized quasipolynomials is developed. The notions of consecution and of finitely based systems of generalized quasipolynomials, that clarify and generalize the ordinary notions of consecution and finitely based $T$-ideals, are introduced. It is shown that systems of homogeneous generalized polynomials (that is, polynomials in variables and in elements of the matrix algebra) are finitely based on this sense. Analogous results are also obtained for systems of ordinary polynomials. A connection with PI-theory is considered. As an application, the representability of a wide class of relative algebras of generalized polynomials is established.