Abstract
Conway semiring-module pairs and iteration semiring-semimodule pairs were shown to provide an axiomatic basis to automata on ω -words in [Bloom, Esik: Iteration Theories, Springer, 1993]. In this paper, we show that two natural classes of semiring-semimodule pairs, the complete and the bi-inductive semiring-semimodule pairs both give rise to iteration semiring-semimodule pairs. Complete semiring-semimodule pairs are defined by infinite sums and products, while a bi-inductive semiring-semimodule pair is an ordered semiring-semimodule pair possessing enough least pre-fixed points and greatest post-fixed points to solve linear inequations. Moreover, we show that when V is idempotent, then a semiring-semimodule pair equipped with a star and an omega operation satisfies the Conway equations (iteration semiring-semimodule pair equations, respectively) if and only if the quemiring associated with (S,V) embeds in a Conway semiring (iteration semiring, respectively).
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Esik, Z., Kuich, W. On Iteration Semiring-Semimodule Pairs. Semigroup Forum 75, 129–159 (2007). https://doi.org/10.1007/s00233-007-0709-7
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DOI: https://doi.org/10.1007/s00233-007-0709-7