The descending chain condition on solution sets for systems of equations in groups

The Ehrenfeucht Conjecture [ 5 ] states that if Μ is a finitely generated free monoid with nonempty subset S , then there is a finite subset T ⊂ S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T . In[ 4 ], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.