Partial Conway and Iteration Semirings

  • Authors:

  • Stephen L. Bloom;Zoltan Ésik;Werner Kuich

  • Affiliations:

  • Dept. of Computer Science, Stevens Institute of Technology, Hoboken, NJ, USA;(Correspd.) GRLMC, Rovira i Virgili University, Tarragona, Spain. E-mail: ze@inf.u-szeged.hu;Inst. for Discrete Mathematics and Geometry, Technical University of Vienna, Austria

  • Venue:
  • Fundamenta Informaticae
  • Year:
  • 2008

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Abstract

A Conway semiring is a semiring S equipped with a unary operation *: S → S, always called 'star', satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as N or N$^{rat}$≪Σ≫ of rational power series of words on Σ with coefficients in N, cannot have a total star operation satisfying the Conway identities. We introduce here partial Conway semirings, which are semirings S which have a star operation defined only on an ideal of S; when the arguments are appropriate, the operation satisfies the above identities. We develop the general theory of partial Conway semirings and prove a Kleene theorem for this generalization.