This theory is an extension of Atomic General Extensional Mereology "AGEM", it adds a primitive total distinctive unary function L, denoting "The label of", and adds the following axioms about it:
L1. Distinctiveness: L(x)=L(y)→x=y
In English: labeling is distinctive, i.e. distinct objects have distinct labels.
L2. Extensiveness: atom(ιx(∄y(x=L(y))))
Where ι is the definite description operator.
In English: there exists only one non-labeling object, and that object is an atom.
L3. Pairing: atom(L([a,b]))
where [a,b] stands for the object composed of just atoms a,b.
In English: The label of a pair of atoms, is an atom.
L4. Perfectness: ∀n[¬atom(Ln(x))]→L(x)=x
In English: If every nthiterative label of a class, is not an atom; then that class labels itself.
Where Ln is defined recursively as:
L1(x)=x
Ln+1(x)=L(Ln(x))
where n is a natural.
To note is that Lnis definable in this theory for every class in this theory. This can be done using fusions of Cartesian products along ways similar to whats presented at this posting.
The idea is to have all objects in this theory being classes, and then define sets as atoms, and define the membership relation ∈ as:
DEFINE (∈):y∈x≡df∃z(x=L(z)∧yPz∧atom(y))
In English: a member of a class is an atom of what's labeled by that class.
This way all classes would be extensional with respect to relation ∈∈. Most of its classes would be mereologically perfect in the sense of having most of their subclasses being parts of them, and also most of their parts being subclasses of them, and also at the same time having all of their members being parts of them. It doesn't use a bottom atom which is mereologically hostile. And it interprets second order arithmetic. It can be extended by adding further axioms about labeling as to interpret any class\set theory, also it can define all species of objects involved in class\set theories, including: Ur-elements, non-extensional classes, sets, classes, proper classes, higher degree proper classes, big sets, all types of non-well founded sets and classes, Multisets, etc.. Also this theory can be used as a foundation of Graph theory and Category theory as well.