Independent Systems of Word Equations: From Ehrenfeucht to Eighteen

Abstract

A system of equations is called independent if it is not equivalent to any of its proper subsystems. We consider the following decades-old question: If we fix the number of variables, then what is the maximal size of an independent system of constant-free word equations? This can be easily answered in the trivial cases of one and two variables, but all other cases remain open, even the three-variable case, where the conjectured answer is as small as three. We survey some historical as well as more recent results related to this question, starting with the one known as Ehrenfeucht’s compactness property: Every infinite system is equivalent to a finite subsystem, and consequently an independent system cannot be infinite. We also discuss several variations and related questions on word equations. Finally, we pay special attention to the following result from 2018: The maximal size of an independent system of three-variable equations is at most 18. This is the first such finite upper bound, but hopefully it will not be the last.