A characterization of abstract families of algebraic power series

Abstract

Given a continuous semiring A and a collection $\mathfrak{H}$ of semiring morphisms mapping the elements of A into finite matrices with entries in A we define $\mathfrak{H}$-closed semirings. These are fully rationally closed semi-rings that are closed under the following operation: each morphism in $\mathfrak{H}$ maps an element of the ,$\mathfrak{H}$-closed semiring on a finite matrix whose entries are again in this $\mathfrak{H}$-closed semiring. $\mathfrak{H}$-closed semirings coincide under certain conditions with abstract families of elements. If they contain only algebraic elements over some A′, A′ $\subseteq$ A, then they are characterized by $\Re \mathfrak{a}\mathfrak{t}$(A′)-algebraic systems of a specific form. The results are then applied to formal power series and formal languages.