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 th
e examples given by Zadori 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.
An $h$-ary relation $r$ on a finite set $A$ is said to be emph{hereditarily rigid} if the unary partial functions on $A$ that preserve $r$ are the subfunctions of the identity map or of constant maps. A family of relations ${mathcal F}$ is said to be
emph{hereditarily strongly rigid} if the partial functions on $A$ that preserve every $r in {mathcal F}$ are the subfunctions of projections or constant functions. In this paper we show that hereditarily rigid relations exist and we give a lower bound on their arities. We also prove that no finite hereditarily strongly rigid families of relations exist and we also construct an infinite hereditarily strongly rigid family of relations.
We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.
