An equivalence relation on a set of words of finite length

In this work, we study several equivalence relations associated to some partitions of sets of finite words. We have results on words over finite fields extending the work of Bacher [R. Bacher, SL2(k)SL2(k) and a subset of words over kk, Europ. J. Combinatorics 23 (2002) 141–147]. Cardinalities of its equivalence classes and explicit relationships between two words are determined. Moreover, we deal with words of finite length over the ring Z/NZZ/NZ where NN is a positive integer. We have arithmetic results parallel to Bacher’s.