On Certain Closure Operators Defined by Families of Semiring Morphisms
✍ Scribed by Georg Karner; Werner Kuich
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 153 KB
- Volume
- 217
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
Given a continuous semiring A and a collection ᑢ of semiring morphisms mapping the elements of A into finite matrices with entries in A we define ᑢ-closed semirings. These are fully rationally closed semirings that are closed under the following operation: each morphism in ᑢ maps an element of the ᑢ-closed semiring on a finite matrix whose entries are again in this ᑢ-closed semiring.
ᑢ-closed semirings coincide under certain conditions with abstract families of elements. If they contain only algebraic elements over some AЈ, AЈ : A, then they Ž . are characterized by ᑬ ᑾ ᒑ AЈ -algebraic systems of a specific form. The results are then applied to formal power series and formal languages. In particular, ᑢ-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.