Abstract
Given a continuous semiring A and a collection H of semiring morphisms mapping the elements of A into finite matrices with entries in A we define H-closed semirings. These are fully rationally closed semirings that are closed under the following operation: each morphism in H maps an element of the H-closed semiring on a finite matrix whose entries are again in this H-closed semiring.H-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 Rat(A′)-algebraic systems of a specific form. The results are then applied to formal power series and formal languages. In particular, H-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.
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