# Definition A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ is - *homogeneous* if for every $p,q$ there is an automorphism $\pi$ s.t. $\pi(p)=q$. - *weakly homogeneous* if for every $p,q$ there is an automorphism $\pi$ such that $\pi(p) \parallel q$. - *locally homogeneous* if for every $p,q$ there are $p^{*}\leq p, q^{*}\leq q$ and an isomorphism $\pi:P\restriction p^{*}\to P\restriction q^{*}$. >[!note] Remark >In many sources weakly homogeneous is called simply homogeneous. # Properties - If $\mathbb{P}$ is locally homogeneous and $G \subseteq \mathbb{P}$ is generic over $V$ then $\mathrm{HOD}^{V[G]}\subseteq \mathrm{HOD}^V(\mathbb{P})$. ^30f0eb - If $\mathbb{P}\in \mathfrak{M}$ is weakly homogeneous and $G \subseteq \mathbb{P}$ is generic over $\mathfrak{M}$ then $\mathrm{HOD}\mathfrak{M}^{\mathfrak{M}[G]}=\mathfrak{M}$ [[Jech - Forcing with trees and ordinal definability]] - If $\mathfrak{N}$ is a model of $\mathrm{ZF}, \mathfrak{M}$ an inner model of $\mathfrak{N}, C \in \mathfrak{M}$ a notion of forcing, $C \in(\mathrm{OD} \mathfrak{M})^{\mathfrak{N}}$, $C$ is homogeneous in $\mathfrak{N}$, then for any $G, C$-generic over $\mathfrak{N}$ $ \mathrm{HOD}^{\mathfrak{M}[G]} \subseteq(\mathrm{HOD} \mathfrak{M})^{\mathfrak{N}[G]} \subseteq(\mathrm{HOD} \mathfrak{M})^{\mathfrak{N}} $ [[Grigorieff - Intermediate Submodels and Generic Extensions in Set Theory]] - A weakly homogeneous forcing satisfies the [[Zero-one law]] (see Kunen pg. 275). # Relations ## Implied by - [[]] ## Implies - [[]] ## Contrasts [[Rigidity]]