Re-partitioning a partitioned set using quotient sets
I hope I've posted this in the right subforum. :)
I've been thinking about a problem for a while, but I've come to a halt. I would be really grateful for insight concerning this. So here goes.
I have a set U consisting of a finite arbitrary number of objects. Each of these objects is a 2-tuple (referred to as T1) where the first element is an arbitrary size finite set, while the second element is a new 2-tuple (referred to as T2). This new 2-tuple consists of two sets - also finite of arbitrary sizes.
The problem is:
I would like to partition the set U using a quotient set. Two objects of U is considered the same (and should reside in the same equivalence class) if the first element of their T1 is equal. ~ (equivalence relation) should be defined by an invariant / morphism that "checks" whether the first element of T1 for two objects is identical (equality of sets). This yields the quotient set U / ~.
The next step is to re-partition the quotient set U / ~ to yield a new quotient set (say V) where the equivalence classes of V are a subset of the equivalence classes of U / ~ (also considering that equivalence classes may "disappear" from U / ~). Thus, V may have fewer equivalence classes than U / ~. Specifically, the elements of each equivalence class of U / ~ should only be members of the "same" equivalence class of V if all objects' T2 tuple have disjoint sets respectively.
Since this is probably not clear just by reading, I've attached a figure illustrating things. It consists of three steps where each transistion (marked by a blue arrow) represents a partitioning.
Concerning the figure: For instance, A is an object of U. It consists of the tuple <Ax, Ay> (referred to as T1). Ax is a set of properties, whereas Ay is a new tuple (T2). T2 has two sets of objects a, b, c,... It can be assumed that all objects of U share the exact same structure.
I apology if my attempt to explain this is somewhat blurry. I is perfectly possible that another notation or structure than what I have used is possible. I have no experience in defining more complex structures than simple sets and tuples.
The initial problem (regardless of my attempt to formalise things) is a set of objects with some meta properties. Two objects are considered equal if their meta properties are the same. A new classification of objects can then be done based on if they contain the same subelements. If someone have a better mathematical structure for expressing this, you are welcome to bring it to the table.
Have a good day!