On Chains of Word Equations and Test Sets

Ehrenfeucht's Conjecture is equivalent with the ascending chain condition for word equations. A related conjecture concerns the descending chain condition for word equations. We show that if this conjecture holds then an arbitrary in nite language L possesses a nite subset F such that the language L F is a disjoint union of its nite test sets. We prove this consequence of the descending chain condition for regular languages while the validity of the descending chain condition in general remains open.