An Equivalence Relation on A Set of Words of Finite Length

TL;DR

This paper investigates equivalence relations on finite-length words, extending existing work over finite fields and rings, and provides explicit class cardinalities and relationships between words.

Contribution

It introduces new equivalence relations on words over finite fields and rings, with explicit class sizes and relationships, extending Bacher's prior work.

Findings

Cardinalities of equivalence classes are determined.

Explicit relationships between words are established.

Arithmetic results over rings parallel those over finite fields.

Abstract

In this work, we study several equivalence relations induced from the partitions of the sets of words of finite length. We have results on words over finite fields extending the work of Bacher (2002, Europ. J. Combinatorics, {\bf 23}, 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 $\mathbb{Z}/N\mathbb{Z}$ where $N$ is a positive integer. We have arithmetic results parallel to Bacher's.