Consider the concept of a module. This can be understood a multisorted algebraic theory $\mathsf{Mod}$ on two sorts, a scalarsort $S$ and a vectorsort $V$. An interpretation of $\mathsf{Mod}$ in $\mathbf{Top}$ assigns to the scalarsort a topological field, to the vectorsort a topological Abelian group, and to the scalar multiplication arrow $S \times V \rightarrow V$ a continuous function satisfying appropriate compatibility constraints.


But we can also imagine *partial* interpretations of $\mathsf{Mod}$ in $\mathbf{Top}.$ E.g. Assign to the scalarsort a topological field $K$, but leave the action and vectorsort unspecified. The result is "a partially interpreted theory" that we might call $K\text{-}\mathsf{Mod}.$ Define that a *model* of $K\text{-}\mathsf{Mod}$ is just a model of $\mathsf{Mod}$ in $\mathbf{Top}$ such that the vectorsort is assigned the topological field $K$. It follows that a model of $K\text{-}\mathsf{Mod}$ is just a topological $K$-module.


Is anyone talking about this kind of thing, where we actually deal with *partial* interpretations of theories?



Is anyone talking about partial interpretations of theories? (Edited)
One common example where this arises is in the consideration of $\omega$-models of set theory. These are precisely the models that interpret their natural numbers as the standard natural numbers $\omega$, or $\mathbb{N}$. Thus, the class of $\omega$-models of a given set theory are essentially the partial interpretations of set theory, where we have fixed a sort for the natural numbers to be interpreted only as the standard natural numbers. 


