Restricted Versions of Hechler Forcing

Question

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.

1. Can we characterize the $\mathcal{F}$ for which $\mathbb{D}(\mathcal{F})$ adds a dominating real?
2. 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.

Answer 1

In case you aren't already aware of this, Jech uses a slightly different poset for Hechler forcing. Specifically, he fixes a family $\mathcal{F} \subseteq \omega^{\omega}$ and lets conditions be of the form $(s, E)$ where $s$ is a finite sequence of Natural numbers and $E$ is a finite subset of $\mathcal{F}$. The conditions are ordered by $(t, F) \leq (s, E)$ exactly when $t$ is an end extension of $s$ and $F \supseteq E$ such that $t(n) > e(n)$ whenever $e \in E$ and $n \in \text{dom}(t) \setminus \text{dom}(s)$ (Jech might have defined it a little differently, not sure). Of course, if we define$\;$$f_E: \omega \rightarrow \omega$ by:

$f_E(n) = \sup\{e(n) \mid e \in E\}$

then the map sending $(s, E)$ to $(s, f_E)$ should bear witness to the forcing equivalence of Jech's characterization of Hechler forcing for $\mathcal{F}$ and your proposed $\mathbb{D}(\mathcal{F})$ when $\mathcal{F}$ is closed under finite suprema.

Now to question (1), you already observed that if you have a function dominating all members of $\mathcal{F}$, then $\mathbb{D}(\mathcal{F})$ cannot add an (eventually) dominating real because for any $n$, it is dense that we stay at or below the dominating function at some later value. When $\mathcal{F}$ is countable, we can easily find such a function by diagonalizing against all of its members. When MA is true, the existence of such a dominating function is extended to families of any size less than $2^{\omega}$. Therefore, as an easy observation for ensuring that $\mathbb{D}(\mathcal{F})$ adds a dominating real, $\mathcal{F}$ must have size at least $\omega_1$ in all cases, and we cannot hope to make any nontrivial blanket characterizations about bounding $\mathcal{F}$'s maximum necessary size in general. Jech actually assumes that $\mathcal{F}$ has size less than $2^{\omega}$ so that he can make the ZFC + MA theorem that the ground model already has a real dominating all functions in $\mathcal{F}$.