The partial clone of linear terms

Generalizing a linear expression over a vector space, we call a term of an arbitrary type $\tau$ linear if its every variable occurs only once. Instead of the usual superposition of terms and of the total many-sorted clone of all terms in the case of linear terms, we define the partial many-sorted superposition operation and the partial many-sorted clone that satisfies the superassociative law as weak identity. The extensions of linear hypersubstitutions are weak endomorphisms of this partial clone. For a variety $V$ of one-sorted total algebras of type $\tau$, we define the partial many-sorted linear clone of $V$ as the partial quotient algebra of the partial many-sorted clone of all linear terms by the set of all linear identities of $V$. We prove then that weak identities of this clone correspond to linear hyperidentities of $V$.