[FOM] Set theories without the extensionality axiom

[ioachim drugus]

Greetings!

I would appreciate if somebody can help me find any reference to an axiomatic set theory without the extensionality axiom. 
Here are merits of such a set theory:
1. This would be a set theory with "multiple universes of discourse", where the "unlimited"
set abstraction {x :F(x)} could be treated as the set abstraction {x in U : F(x)}, limited by a universe of discourse on which we are focused (as many authors,  I am using the reserved word "in" as a denotation of the membership relationship in plain text).   
2. The entities studied by such a theory would be multi-identity. This would be a theory of multi-sets, where the modifier "multi-" refers not only to the elements of a set which may encounter in many copies (replicas), but also to the set itself - it is allowed to also be present in many copies. Also, the modifier "multi-" applies to the atoms (urelements) - they can also be present in multiple copies. 

Thank you,
Ioachim Drugus,
Main Software Architect,
SemanticSoft, Inc.