$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ has property $\mu$-linked and $\mu$-centered.

Question

Let $\mathbb{P}$ be a poset and $\mathbb{\dot{Q}}$ be a $\mathbb{P}$-name of a poset.

If $\phi(\mathbb{P})$ and $\Vdash_{\mathbb{P}} \phi(\mathbb{\dot{Q}})$, then $\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ where $\phi(x)$ is one of the following properties for $\mu$ infinite cardinal:

$(i)$ $x$ is $\mu$-linked.

$(ii)$ $x$ is $\mu$-centered.

Any ideas.

Thanks.

Answer 1

I will only address (i), as (ii) is a slight modification of this.

(i) Let $\mathbb P = \bigcup_{\alpha < \mu} L_\alpha$, $L_\alpha$ linked for each $\alpha \in \mu$ and fix a $\mathbb P$-Name $\tau$ such that

$ 1_{\mathbb P} \Vdash \left\{ \begin{array} \ \tau \text{ is a function with domain } \mu, \\ \forall \gamma \in \mu: \tau(\gamma) \subseteq \dot{ \mathbb Q}, \\ \forall x \in \dot{ \mathbb Q} \exists \gamma \in \mu: x \in \tau(\gamma), \\ \forall \gamma \in \mu \forall x,y \ (x,y \in \tau(\gamma) \rightarrow \exists z \in \dot {\mathbb Q}: z \dot \le x \wedge z \dot \le y) \end{array} \right.$

For $\alpha, \beta, \gamma \in \mu$ let

$$ L_{(\alpha,\beta,\gamma)}^* = \{ (p,\dot q) \in L_\alpha \times \dot {\mathbb Q} \mid p \Vdash_{\mathbb P} \dot q \in \dot {\mathbb Q}, \exists r \in L_\beta, r \le_{\mathbb P} p, r \Vdash_{\mathbb P} \dot q \in \tau(\gamma)\} $$

• $L_{(\alpha,\beta,\gamma)}^*$ is linked:

Let $(p_0,\dot q_0), (p_1,\dot q_1) \in L_{(\alpha,\beta,\gamma)}^*$ and fix $r_0,r_1 \in L_\beta$ s.t. $$ r_i \Vdash_{\mathbb P} \dot q_i \in \tau(\gamma) \text{, for } i=0,1 $$ As $L_{\beta}$ is linked, we may fix $r \le_{\mathbb P} r_0,r_1$ and (as $1_{\mathbb P} \Vdash_{\mathbb P} \tau(\gamma) \text{ is linked}$) $\dot q \in dom(\dot {\mathbb Q})$ s.t. $$ r \Vdash_{\mathbb P} \dot q \dot \le \dot q_i \text{, for } i=0,1 $$ Now $(r,\dot q) \le^* (p_0, \dot q_0), (p_1, \dot q_1)$ and $L_{(\alpha, \beta, \gamma)}^*$ is shown to be linked.

• $\mathbb P \ast \dot {\mathbb Q} = \bigcup\{L_{(\alpha, \beta, \gamma)}^* \mid \alpha,\beta,\gamma \in \mu\}$:

"$\supseteq$": $\checkmark$

"$\subseteq$": Let $(p,\dot q) \in \mathbb P \ast \dot {\mathbb Q}$, say $p \in L_\alpha$. Fix $r \le_{\mathbb P} p$, $\gamma \in \mu$ s.t. $$ r \Vdash_{\mathbb P} \dot q \in \tau(\gamma) $$ Let $\beta \in \mu$ be s.t. $r \in L_\beta$. Then $(p,\dot q) \in L_{(\alpha, \beta, \gamma)}^*$ and the claim follows.