## Background

This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing.

In his question, Justin considers a relation $\lhd$ on partial orders (defined below as $\lhd_1$) which begins to capture the notion of one partial order being "stronger" than another. He asks a natural question, namely whether this relation satisfies antisymmetry -- that is, if $\mathbb{Q}\lhd\mathbb{P}$ and $\mathbb{P}\lhd\mathbb{Q}$, does it follow that $\mathbb{P} \sim \mathbb{Q}$? Here, $\sim$ means *forcing equivalent*: two partial orders $\mathbb{P}$ and $\mathbb{Q}$ are forcing equivalent if they give rise to the same generic extensions, that is, every extension $V[G]$ by $\mathbb{P}$ can be realized as an extension $V[H]$ by $\mathbb{Q}$, and vice versa (in Joel Hamkins' terminology, we could say they are forcing equivalent if they give us access to the same neighborhood of the multiverse).

Unfortunately the given relation does not satisfy antisymmetry. In fact, the counterexamples given in the responses to the question demonstrate that $\lhd$ falls short of capturing the idea of one partial order being stronger than another. I'm trying to find the "right version" of $\lhd$ -- a relation on partial orders $\mathbb{Q} \lhd \mathbb{P}$ that captures our intuitive notion of '$\mathbb{P}$ *is stronger than* $\mathbb{Q}$'. A good test criteria is whether the relation satisfies antisymmetry. I will consider a number of (I hope) natural strengthenings of the $\lhd$ relation (or variations -- it's not clear that all of them are strengthenings).

Consider the following relations on partial orders:

$\mathbb{Q} \lhd_1 \mathbb{P}$:

*Every $\mathbb{P}$ extension contains a $\mathbb{Q}$ extension.*$\mathbb{Q} \lhd_1 \mathbb{P}$ if and only if whenever $V[G]$ is a forcing extension by $\mathbb{P}$, there is an $H\in V[G]$ generic for $\mathbb{Q}$ with $V[H] \subseteq V[G]$. (this is the original definition from Justin's post).$\mathbb{Q} \lhd_2 \mathbb{P}$:

*Every $\mathbb{P}$ extension contains a $\mathbb{Q}$ extension meeting an arbitrary condition.*$\mathbb{Q} \lhd_2 \mathbb{P}$ if and only if whenever $V[G]$ is an extension by $\mathbb{P}$, and $q \in \mathbb{Q}$, there is $H$ in $V[G]$ generic for $\mathbb{Q}$ with $q \in H$.$\mathbb{Q} \lhd_3 \mathbb{P}$:

*Every $\mathbb{Q}$ extension is contained in a $\mathbb{P}$ extension.*$\mathbb{Q} \lhd_3 \mathbb{P}$ if and only if whenever $V[H]$ is an extension by $\mathbb{Q}$, there is an extension $V[G]$ by $\mathbb{P}$ such that $V[H] \subseteq V[G]$.$\mathbb{Q} \lhd_4 \mathbb{P}$: $\mathbb{Q}$

*embeds into*$\mathbb{P}$. $\mathbb{Q} \lhd_4 \mathbb{P}$ if and only if there is a complete embedding of (the Boolean algebra of) $\mathbb{Q}$ into (the Boolean algebra of) $\mathbb{P}$.$\mathbb{Q} \lhd_5 \mathbb{P}$:

*Every $\mathbb{Q}$ extension is equal to a $\mathbb{P}$ extension.*$\mathbb{Q} \lhd_5 \mathbb{P}$ if and only if whenever $V[H]$ is an extension by $\mathbb{Q}$, there is an extension $V[G]$ by $\mathbb{P}$ such that $V[H] = V[G]$.

Please note that I have ignored serious metamathematical considerations in giving these definitions, but I believe each of them has a first-order definition.

Question.Which of the relations above satisfy antisymmetry? That is, for which of the relations above do we have $\mathbb{Q}\lhd\mathbb{P}$ and $\mathbb{P}\lhd\mathbb{Q}$ implies $\mathbb{P}$ is forcing equivalent to $\mathbb{Q}$?

From the responses to Justin's question it is clear that $\lhd_1$ does not satisfy antisymmetry. On the other hand, $\lhd_5$ does satisfy antisymmetry, but is extremely restrictive in terms of the partial orders that it can compare (for example, $Add(\omega,1)$ and $Add(\omega,1)\times Add(\omega_1,1)$ are incomparable by $\lhd_5$, although the latter is apparently more powerful than the former).

At the risk of violating the suggested guidelines for keeping questions focussed and specific, I'll include two followup questions -- feel free to ignore.

Question 2.What are the relative strengths of these relations (which ones imply which others)?

For example, we (trivially) have $\mathbb{Q} \lhd_5 \mathbb{P}$ implies $\mathbb{Q} \lhd_3 \mathbb{P}$, and $\mathbb{Q} \lhd_2 \mathbb{P}$ implies $\mathbb{Q} \lhd_1 \mathbb{P}$.

Question 3.What is the "right" notion of $\lhd$ for partial orders, that captures our intuitive idea of one partial order being "stronger" or "more effective" than another?

This is "too vague", but it's the question that motivated those above.

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