1. Axiomatizing rational power series over natural numbers
- Author
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Stephen L. Bloom and Zoltán Ésik
- Subjects
Power series ,Class (set theory) ,Mathematics::Commutative Algebra ,Group (mathematics) ,Natural number ,Characterization (mathematics) ,Theoretical Computer Science ,Computer Science Applications ,Combinatorics ,Regular language ,Computational Theory and Mathematics ,Calculus ,Order (group theory) ,Variety (universal algebra) ,Mathematics ,Information Systems - Abstract
Iteration semi-rings are Conway semi-rings satisfying Conway's group identities. We show that the semi-rings N^r^a^t > of rational power series with coefficients in the semi-ring N of natural numbers are the free partial iteration semi-rings. Moreover, we characterize the semi-rings N"~"^"r"^"a"^"t > as the free semi-rings in the variety of iteration semi-rings defined by three additional simple identities, where N"~ is the completion of N obtained by adding a point of infinity. We also show that this latter variety coincides with the variety generated by the complete, or continuous semirings. As a consequence of these results, we obtain that the semi-rings N"~^r^a^t >, equipped with the sum order, are free in the class of symmetric inductive ^*-semi-rings. This characterization corresponds to Kozen's axiomatization of regular languages.
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