A Language Hierarchy of Binary Relations

TL;DR

This paper investigates a hierarchy of binary relations on words, classifying them based on language classes and perspectives, with applications to monoid word problems.

Contribution

It introduces a new hierarchy of binary relation classes on words, linked to language classes and monoid word problems, providing a structured framework.

Findings

Established a hierarchy of relation classes

Provided examples distinguishing classes

Connected relation classes to monoid word problems

Abstract

Motivated by the study of word problems of monoids, we explore two ways of viewing binary relations on $A^*$ as languages. We exhibit a hierarchy of classes of binary relations on $A^*$, according to the class of languages the relation belongs to and the chosen viewpoint. We give examples of word problems of monoids distinguishing the various classes.