Q:

Asking for refs: formalisms that admit {x}={{x}}

I have some interest in alternative formulations of set theory which blatantly admit {x}={{x}}. What gave rise to such interest is the following. Take the set equation A={A}. The RHS dictate that {A} contains one element. So since A={A}, A is also of cardinality 1. We therefore get {x}={{x}} which has no solution. But if we admitted that {x}={{x}}, we could have a solution: A can be any set of a single element. Under such formalism, therefore, set equations become more "algebrically closed".
I'll be thankful if anyone can point me to such existing formalisms of set theory as I didn't find any.
Edit:
we will need to give up the regularity axiom
https://en.wikipedia.org/wiki/Axiom_of_regularity#No_set_is_an_element_of_itself
but that's the edge of my knowledge

A:

There are well-studied set theories where the equation $x=\{x\}$ has a solution (and therefore also $\{x\}=\{\{x\}\}$, of course). Such a solution is sometimes called a Quine atom.
Look for non-well-founded set theory, and for the anti-foundation axiom in particular.

On the other hand, having $\forall x.\{x\}=\{\{x\}\}$ would require major changes to our conception of sets, and I'm not sure the result would be recognizable as set theory at all.
In particular, the notation $\{x\}$ is usually taken to mean "a set whose only element is $x$", so if $\{x\}=\{\{x\}\}$, then we have one set whose only element is simultaneously $x$ and $\{x\}$. This means that $x=\{x\}$, or again that the only element of $x$ is $x$ itself.
So in such a theory, if you want any set that contains two different things (or just something different from itself), you would need to give up the usual interpretation of $\{\cdots\}$ in terms of $\in$.
Trying to work out what you want to put instead, you might end up reinventing some form of mereology.

