Abstract
We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving ultrafilter equations characterising B and then projecting these to obtain profinite equations characterising B∩Reg. This suffices to establish the decidability of the membership problem for B∩Reg.
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