The Generalized Rank of Trace Languages
The notion of rank of a language with respect to an independence alphabet is generalized from concatenations of two words to an arbitrary fixed number of words. It is proved that in the case of free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular languages are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular languages and rational series over the min-plus semiring, it is shown that already for free products of free commutative monoids, rank sequences need not be eventually periodic.
KeywordsTrace language Rank Regular language Rational series Tropical semiring
We are grateful to Jacques Sakarovitch and Sylvain Lombardy for pointing us to the result of Simon .
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