On Certain Closure Operators Defined by Families of Semiring Morphisms

Given a continuous semiring A and a collection of semiring morphisms mapping the elements of A into finite matrices with entries in A we define -closed semirings. These are fully rationally closed semirings that are closed under the following operation: each morphism in maps an element of the -closed semiring on a finite matrix whose entries are again in this -closed semiring. -closed semirings coincide under certain conditions with abstract families of elements. If they contain only algebraic elements over some A′, A′ A, then they are characterized by (A′)-algebraic systems of a specific form. The results are then applied to formal power series and formal languages. In particular, -closed semirings are set in relation to abstract families of elements, power series, and languages. The results are strong “normal forms” for abstract families of power series and languages.