Partial Conway and Iteration Semiring-Semimodule Pairs
A Conway semiring is a semiring S equipped with a unary operation *:S → S, called “star”, satisfying the sum star and product star identities. A Conway semiring-semimodule pair consists of a Conway semiring S and a left S-semimodule V together with a function ω : S → V, called “omega power”, subject to the sum omega and product omega identities. A Kleene type theorem holds in all Conway semiring-semimodule pairs that can be instantiated to give the equivalence of Büchi automata and regular languages over ω-words. However, sometimes the star and omega power operations cannot be defined in an appropriate manner on the whole semiring S. To handle this situation, we introduce partial Conway semiring-semimodule pairs and develop their basic theory in connection with automata. We prove a Kleene theorem, applicable to all partial Conway semiring-semimodule pairs.
KeywordsMatrix Theory Matricial Theory Regular Language Distinguished Ideal Rational Element
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