On certain closure operators defined by families of semiring morphisms

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' subset of or equal to 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. (C) 1999 Academic Press.