Boolean closure and unambiguity of rational sets
We consider three properties that can be verified by the rational subsets of a monoid M: to coincide with the recognizable subsets of M, to coincide with the unambiguous rational subsets of M, to form a boolean algebra. We study what connections exist between these properties. We build a monoid in which rational subsets are recognizable (and thus form a boolean algebra), but are not all unambiguous and a monoid in which rational subsets are unambiguous but do not form a boolean algebra. We show that the class of monoids the rational subsets of which are recognizable and unambiguous is not closed by finitely generated submonoids.
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