Idempotent bonding relations are nontrivial if and only if they satisfy condition Gamma

TL;DR

This paper characterizes nontrivial idempotent bonding relations on compact Hausdorff spaces as those satisfying a specific condition, Gamma, improving previous results in the field.

Contribution

It provides a precise characterization of nontrivial idempotent bonding relations using condition Gamma, advancing the understanding of their structure.

Findings

Nontrivial idempotent bonding relations satisfy condition Gamma.

Condition Gamma involves the existence of distinct points with specific relation properties.

The result applies to compact Hausdorff spaces, refining earlier characterizations.

Abstract

A relation $f\subseteq X^2$ satisfies condition $\Gamma$ if there exist distinct $x,y\in X$ with $\langle x,x\rangle ,\langle x,y\rangle ,\langle y,y\rangle \in f$. The authors improve a previous result by characterizing nontrivial idempotent bonding relations on compact Hausdorff spaces as those satisfying condition $\Gamma$.