Hereditarily rigid 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.