# Definition >[!info] Definition >$\mathbb{P}$ poset, $\mathbb{P}\in M\prec H_{\theta}$. >$q$ is *$(M,\mathbb{P})$-semigeneric* if $\forall r\leq q \forall D\in M$ dense $\exists u\in D$ s.t. $Hull(M,\{u\})\cap \omega_{1}=M\cap \omega_{1}$ s.t. $u\parallel r$ >Where $Hull(M,\{u\})=\{f(u)\mid f\in M, u\in \mathrm{dom}(f)\}$ > Equivalently: $q\Vdash M[\dot{G}\cap\omega_{1}]=M\cap\omega_{1}$ > A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ is *semiproper* for $M$ if $\forall p\in M\cap \mathbb{P}\exists q\leq p$ which is $(M,\mathbb{P})$-semigeneric # Properties # Relations ## Implied by - [[Proper forcing]] ## Implies - [[]] # Preservation