In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the existence of nontrivial automorphisms and nontrivial elementary embeddings of the universe in models of set theory without Foundation take the following form:

{$id_{V}$}$\subseteq$$Aut(V)$$\subseteq$$Eem(V)$

where $id_{V}$ is just the identity mapping from $V$ to $V$, $Aut(V)$ are the automorphisms from $V$ to $V$,and $Eem(V)$ are the elementary embeddings from $V$ to $V$.

In fact, they prove that there are models of $ZFC^{-f}$ that realize each of these four separating refinements of {$id_{V}$}$\subseteq$$Aut(V)$$\subseteq$$Eem(V)$:

i). {$id_{V}$}=$Aut(V)$=$Eem(V)$

ii). {$id_{V}$}$\subsetneq$$Aut(V)$=$Eem(V)$

iii).{$id_{V}$}=$Aut(V)$$\subsetneq$$Eem(V)$

iv). {$id_{V}$}$\subsetneq$$Aut(V)$$\subsetneq$$Eem(V)$

My question is simply this:

Are there models of $NGB+{\lnot}AC$ that realize each of the four separating refinements?

weaker. It says that there exists a class $A$ such that for every relation $R$, $(A,R)$ is not rigid. This does not in any way contradict the fact that there is a different class, namely $V$, which does carry a rigid relation. Yair’s answer says what it says: if it is consistent with NBG that there is a nontrivial elementary embedding $V\to V$, then the existence of such an elementary embedding is also consistent with there being a set that does not carry a rigid relation. $\endgroup$ – Emil Jeřábek supports Monica Apr 12 '15 at 14:55externalautomorphisms of the model for which replacement needn’t hold. This is a whole different game.Everyfirst-order model has elementary extensions with lots of such automorphisms. $\endgroup$ – Emil Jeřábek supports Monica Apr 24 '15 at 12:12