# Definition A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ is *rigid* if it has no non-trivial automorphisms Let $B^*$ denote these elements of a complete Boolean algebra $B$ that are left fixed by every automorphism of $B$. We call $B^*$ the *rigid part* of $B$. ^rigid-part # Properties - Vôpenka: If $G$ is $B$-generic over $\mathfrak{M}$, then $(\mathrm{HOD} \mathfrak{M})^{\mathfrak{M}[G]}=\mathfrak{M}\left[G \cap B^*\right]$. - $\mathrm{HOD}\mathfrak{M}^{\mathfrak{M}[G]}=\mathfrak{M}[G]$ [[Jech - Forcing with trees and ordinal definability]] # Relations ## Implied by - [[]] ## Implies - [[]] ## Contrasts [[Homogeneity]] # Preservation