Definition (1): An $\mathcal{L}$ - structure $\mathcal{M}$ called "rigid" iff there is no non-trivial automorphism on $\mathcal{M}$.
Definition (2): An $\mathcal{L}$ - structure $\mathcal{M}$ called "non-rigid" iff it is not a rigid model.
Definition (3): An $\mathcal{L}$ - structure $\mathcal{M}$ called "flexible" iff $\mathcal{M}$ has the "maximum" number of automorphisms.(For structures with a set as their domain of discourse this means $|Aut(\mathcal{M})|=2^{|\mathcal{M}|}$ and for the structures with a proper class domain we "define" this notion to be $|Aut(\mathcal{M})|=|Ord|$ which is an abbreviation for "there are class-many automorphisms")
Definition (4): By replacing the notion of "automorphism" with "self elementary embedding" one can define the "pseudo rigid", "pseudo non-rigid" and "pseudo flexible" structures in the similar way on definitions (1), (2) and (3).
There are many (set and proper class) rigid models in set theory and model theory (like $\mathbb{R}$ as an ordered field or $V$ as an $\in$-model of $ZFC$). But even there are many "origamic" model construction methods which can change the nature of a model "softely". Ultrapower is one of these operators. Intuitionally the possibility of occuring a non-trivial self-similarity in a rigid model increases with corrugating it by larger ultrapowers. Now there are some questions here:
Question (1): Let $\mathcal{M}$ be a set-kind rigid $\mathcal{L}$ - structure. Are there an index set $I$ and an ultrafilter $\mathcal{F}$ on it such that $\Pi_{\mathcal{F}}\mathcal{M}$ be a "non-rigid" model ($|Aut(\Pi_{\mathcal{F}}\mathcal{M})|>1$)?
Question (2): Let $\mathcal{M}$ be a set-kind rigid $\mathcal{L}$ - structure. Are there an index set $I$ and an ultrafilter $\mathcal{F}$ on it such that $\Pi_{\mathcal{F}}\mathcal{M}$ be a "flexible" model ($|Aut(\Pi_{\mathcal{F}}\mathcal{M})|=2^{|\Pi_{\mathcal{F}}\mathcal{M}|}$)?
Question (3): What are the answers of above questions for an arbitary rigid proper class model (like $\langle V,\in\rangle$)?
Remark (1): The notion of the ultrapower for a proper class model can be defined by "Scott's trick" to produce another proper class model.
Even there are some questions about possible resolving of "pseudo rigidity" problem of $\langle V,\in \rangle$ by ultraproducts:
Question (4): Are there large cardinal axioms like $A$ and $B$ such that:
(a) $ZFC+A\Longrightarrow \exists M~\exists j:\langle V,\in \rangle \longrightarrow \langle M,\in \rangle~~;~~\langle M,\in \rangle~is~a~"pseudo~non-rigid"~inner~model~of~ZFC$ \$~~\wedge~~j~is~a~non-trivial~elementary~embedding$
(b) $ZFC+B\Longrightarrow \exists M~\exists j:\langle V,\in \rangle \longrightarrow \langle M,\in \rangle~~;~~\langle M,\in \rangle~is~a~"pseudo~flexible"~inner~model~of~ZFC$ \$~~\wedge~~j~is~a~non-trivial~elementary~embedding$
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