Restricted Versions of Hechler Forcing

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.

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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 $\mathcal{F}$.

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This of course is only a partial answer to (1). – Jason Mar 9 '11 at 8:58
The explanation "because, for any $n$, it is dense that we stay at or below the dominating function at some later value" (in the last paragraph of your answer) seems to explain only why the generic real given by the forcing isn't dominating, not why no dominating real is added. Fortunately, though, the rest of that paragraph can be based on the general fact that, if a notion of forcing has cardinality smaller than the dominating number $\mathfrak d$, then the forcing extension contains no dominating real. – Andreas Blass Mar 9 '11 at 14:47
I was not aware that Jech's book used that forcing. It seems strange since he doesn't seem to make any actual use of the added restriction. – Justin Palumbo Mar 9 '11 at 16:23
@Andreas: Yes, that's an important observation here because for more complex posets we can surreptitiously code a dominating real in the generic one; @Justin: Yeah, I've only seen Hechler forcing mentioned (almost as an aside) in the context of applications of MA and as an exercise in Jech. He seems to use such a family only to limit its size; we obviously cannot find a real in a ZFC + MA model dominating all of its own reals. But he also could've just done without the condition and talked about only dominating fewer than $2^{\omega}$ many so maybe he was just making the forcing more general. – Jason Mar 10 '11 at 2:51