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.
Let A be a finite non-singleton set. For |A|=2 we show that the partial clone consisting of all selfdual monotone partial functions on A is not finitely generated, while it is the intersection of two finitely generated maximal partial clones on A. Mo
reover for |A| >= 3 we show that there are pairs of finitely generated maximal partial clones whose intersection is a non-finitely generated partial clone on A.
