A compactness property of the k-abelian monoids

Abstract

The k-abelian equivalence of words, counting the numbers of occurrences of factors of length at most k, has been analyzed in recent years from several different directions. We continue this analysis. The k-abelian equivalence classes are known to constitute a monoid. Hence, equations over these monoids are well defined. We show that these monoids satisfy a compactness property: each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems.We give two proofs for this compactness result. One is based the fact that the monoid can be embedded into the (multiplicative) monoid of matrices, and the other directly on linear algebra. The former method allows the application of Hilbert's basis theorem. The latter one, in turn, allows to conclude an upper bound for the size of the finite subsystem.