The Monoid Of Binary Relations On A Set Of Size Four Has Infinite Representation Type

TL;DR

This paper investigates the representation type of monoids related to binary relations and generalized correspondences, providing new results and conditions for infinite representation type.

Contribution

It extends prior work on transformation monoids to binary relations and introduces a criterion for infinite representation type in monoids.

Findings

The monoid of binary relations on a set of size four has infinite representation type.

A partial result is provided for the monoid of Λ-generalized correspondences.

A sufficient condition for a monoid to have infinite representation type is established.

Abstract

The problem of determining the representation type of the full transformation monoid was resolved by Ponizovskii, Putcha, and Ringel. In this paper, we present a similar result for the monoid of binary relations, a partial result for the monoid of $\Lambda$-generalized correspondences, and a sufficient condition for a monoid to have infinite representation type.