On mappings of terms determined by hypersubstitutions

The extensions of hypersubstitutions are mappings on the set of all terms. In the present paper we characterize all hypersubstitutions which provide bijections on the set of all terms. The set of all such hypersubstitutions forms a monoid.
On the other hand, one can modify each hypersubstitution to any mapping on the set of terms. For this we can consider mappings $\rho$ from the set of all hypersubstitutions into the set of all mappings on the set of all terms. If for each hypersubstitution $\sigma$ the application of $\rho(\sigma )$ to any identity in a given variety $V$ is again an identity in $V$, so that variety is called $\rho$-solid. The concept of a $\rho$-solid variety generalizes the concept of a solid variety. In the present paper, we determine all $\rho$-solid varieties of semigroups for particular mappings $\rho$.