Given $\mathcal{F}\subseteq\omega^\omega$, let $\mathbb{D}(\mathcal{F})$ denote the forcing which is much like Hechler forcing, but now in the conditions $\langle s,f\rangle$ we require that $f$ come from $\mathcal{F}$. (Thus taking $\mathcal{F}$ to be $\omega^\omega$ just gives us the usual Hechler forcing). Has this notion of forcing been used or studied before?

Beyond that I have two main questions.

- Can we characterize the $\mathcal{F}$ for which $\mathbb{D}(\mathcal{F})$ adds a dominating real?
- Can we characterize the $\mathcal{F}$ for which Cohen forcing will add a filter which is $\mathbb{D}(\mathcal{F})$-generic?

I think probably one should assume that for any $f,g\in\mathcal{F}$ there is $h\in\mathcal{F}$ such that $f,g\leq h$; this guarantees that two conditions with the same stem are compatible and thus that the forcing is ccc. Also let us assume that for every $f\in\mathcal{F}$ there is $g\in\mathcal{F}$ with $f< g$, so that $\mathcal{F}$ has no maximal elements. Finally, it does us no harm to assume that $\mathcal{F}$ is closed downwards under $\leq$ since closing it downward will not affect the forcing.

On question 1: I can show that if $\mathcal{F}$ is bounded above by some $f$, then $\mathbb{D}(\mathcal{F})$ does not add a dominating real. And if there is some infinite $a\subseteq\omega$ such that $\{f\upharpoonright a:f\in\mathcal{F}\}$ dominates the functions in $\omega^a$ then it does add a dominating real. Perhaps that gives the appropriate characterization?

On question 2: Certainly if $\mathcal{F}$ is countable then adding a Cohen real adds a $\mathbb{D}(\mathcal{F})$-generic. Is that the only situation? I suspect so but then again it seems plausible it might also be able to happen for certain bounded $\mathcal{F}$. However, I can show that adding a Cohen real does not add a generic in the following situation: fix some $h\in\omega^\omega$ with $\lim h(n)=\infty$ and let $\mathcal{F}$ be all functions $f$ such that $\lim h(n)-f(n)=\infty$

Motivation: There is an intuition for Hechler forcing that it consists of two parts: a Cohen part and a dominating part. For example, there is an old result of Truss that if $d$ dominates $V$ and $c$ is Cohen-generic over $V[d]$ then $d+c$ is Hechler-generic over V. This question is to explore that intuition a little further.