# Strong 2.t and Strong 3.t Transformations for Strong M-equivalence

## TL;DR

This paper introduces strong 2.t and 3.t transformations to analyze strong M-equivalence of words, providing structural insights and addressing limitations in characterizing equivalence classes.

## Contribution

It proposes new order-independent transformations for studying strong M-equivalence and analyzes their properties and limitations.

## Key findings

- Introduces strong 2.t and 3.t transformations.
- Provides structural characterization of irreducible strong 2.2 transformations.
- Addresses limitations of transformations in characterizing strong M-equivalence.

## Abstract

Parikh matrices have been extensively investigated due to their usefulness in studying subword occurrences in words. Due to the dependency of Parikh matrices on the ordering of the alphabet, strong M-equivalence was proposed as an order-independent alternative to M-equivalence in studying words possessing the same Parikh matrix. This paper introduces and studies the notions of strong 2.t and strong 3.t transformations in determining when two ternary words are strongly M-equivalent. The irreducibility of strong 2.t transformations are then scrutinized, exemplified by a structural characterization of irreducible strong 2.2 transformations. The common limitation of these transformations in characterizing strong M-equivalence is then addressed.