Equations over the k-Binomial Monoids

Abstract

Two finite words u and v are k-binomially equivalent if each word of length at most k appears equally many times in u and v as a subword, or scattered factor. We consider equations in the so-called k-binomial monoid defined by the k-binomial equivalence relation on words. We remark that the k-binomial monoid possesses the compactness property, namely, any system of equations has a finite equivalent subsystem. We further show an upper bound, depending on k and the size of the underlying alphabet, on the number of equations in such a finite subsystem. We further consider commutativity and conjugacy in the k-binomial monoids. We characterise 2-binomial conjugacy and 2-binomial commutativity. We also obtain partial results on k-binomial commutativity for \(k > 2\).