Semirigid systems of three equivalence relations

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.