Abstract
A variation of first-order logic with variables for exponents is developed to solve some problems in the setting of recognizable languages on the free monoid, accommodating operators such as product, bounded shuffle and reversion. Restricting the operators to powers and product, analogous results are obtained for recognizable languages of an arbitrary finitely generated monoid M, in particular for a free inverse monoid of finite rank. As a consequence, it is shown to be decidable whether or not a recognizable subset of M is pure or p-pure.
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