Semirigid systems of three equivalence relations

TL;DR

This paper constructs semirigid systems of three equivalence relations on sets, extending previous examples and showing their existence on various infinite and finite sets, with implications for algebraic structures.

Contribution

The authors construct new semirigid systems of three equivalence relations, generalizing prior examples and extending their existence to infinite cardinalities.

Findings

Existence of semirigid systems on sets of continuum size

Extension of Zádori's 1983 examples

Construction applicable to infinite cardinalities

Abstract

A system $\mathcal M$ of equivalence relations on a set $E$ is \emph{semirigid} if only the identity and constant functions preserve all members of $\mathcal M$. We construct semirigid systems of three equivalence relations. Our construction leads to the examples given by Z\'adori in 1983 and to many others and also extends to some infinite cardinalities. As a consequence, we show that on every set of at most continuum cardinality distinct from $2$ and $4$ there exists a semirigid system of three equivalence relations.