This paper continues the algebraic theory of Esik, Kuich [9] on semiring-semimodule pairs and quemirings that is applicable to languages that contain finite and infinite words. The main advantage is that we get rid of the idempotency assumption for the semimodule needed at several places in Esik, Kuich. Additionally, we consider linear systems as a generalization of rightlinear grammars. Moreover, we develop an algorithm that constructs, for a given finite automaton, an equivalent one without ε-moves.
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