The partial clone of linear formulas

A term $t$ is linear if no variable occurs more than once in $t$. An identity $s\approx t$ is said to be linear if $s$ and $t$ are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra ${\operatorname{Formclone}}_{\operatorname{lin}}(\tau,\tau')$. This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type $\tau$ to the linear hypersubstitutions of type $(\tau,\tau')$ for algebraic systems. The extensions of linear hypersubstitutions of type $\tau,\tau'$ send linear formulas to linear formulas, presenting weak endomorphisms of ${\operatorname{Formclone}}_{\operatorname{lin}}(\tau,\tau')$.