Cohen Generic Structures with Functions

Suppose $\mathscr{L}^-\subseteq \mathscr{L}$ are languages where $\mathscr{L} \setminus\mathscr{L}^-$ is relational. Additionally, let $\mathbf{K}$ be a strong $\textrm{Fra\"iss\'e}$ class in $\mathscr{L}$. We consider the partial ordering, under substructure, of those elements in $\mathbf{K}$ whose reduct to $\mathscr{L}^-$ are substructures of a fixed $\mathscr{L}^-$-structure $\mathcal{M}^-$. In this paper, we establish that, under general conditions, this partial order satisfies the $|\mathcal{M}^-|$-chain condition. Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse\ limit of $\mathbf{K}$, provided $\mathcal{M}^-$ satisfies the theory of$\textrm{Fra\"iss\'e}$ limit of its age. We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups.