For a class of forcing notions $\mathcal{P}$, denote: $\mathbf{M A}(\mathcal{P})$: If $P$ is a forcing notion in $\mathcal{P}$ and if $\left\{D_\alpha\right.$ : $\left.\alpha<\omega_1\right\}$ are dense (or predense) subsets of $P$, then there exists a filter $G$ on $P$ that meets all the $D_\alpha$. Bounded $\mathbf{M A}(\mathcal{P})$: If $P$ is a forcing notion in $\mathcal{P}$ and if $\left\{D_\alpha: \alpha<\omega_1\right\}$ are predense subsets of $P$ such that $\left|D_\alpha\right| \leq \aleph_1$ for all $\alpha$, then there exists a filter $G$ on $P$ that meets all the $D_\alpha$. $\mathbf{M A}^{+}(\mathcal{P})$: If $P$ is a forcing notion in $\mathcal{P}$, if $\left\{D_\alpha: \alpha<\omega_1\right\}$ are dense (or predense) subsets of $P$ and if $\dot{\tau}$ is a name of a subset of $\omega_1$ such that $\Vdash \dot{\tau}$ is stationary, then there exists a filter $G$ on $P$ that meets all the $D_\alpha$, and $\dot{\tau}^G=\{\alpha: \exists p \in G p \Vdash \alpha \in \dot{\tau}\}$ is stationary. $\mathbf{M A}^{+\beta}(\mathcal{P})$: If $P$ is a forcing notion in $\mathcal{P}$, if $\left\{D_\alpha: \alpha<\omega_1\right\}$ are dense (or predense) subsets of $P$ andand $\left\langle\tau_\alpha: \alpha<\beta\right\rangle$ is a sequence of $P$-names for stationary subsets of $\omega_1$, there is a filter $G \subseteq P$ such that $G \cap D_\alpha$ is nonempty for each $\alpha<\omega_1$ and $\left\{\gamma<\omega_1 \mid \exists p \in G p \Vdash \check{\gamma} \in \tau_\alpha\right\}$ is stationary for each $\alpha<\beta$. >[!info] Source >see Jech and Handbook - Forcing over Models of Determinacy >[!example] > [[Martin's Axiom]] = $\mathrm{MA}(ccc)$ > [[Proper forcing axiom]] = $\mathrm{MA}(\text{proper})$ > [[Martin's maximum]] - $\mathrm{MA}(\text{preserves stationary in }\omega_{1})$