What Mathematical Branch Studies Relations Between Multisets and Normal Sets?

[Organic]

Let x be a general notation for a singleton.

When a finite collection of singletons have the same color, we mean that all singletons are identical, or have the maximum symmetry-degree.

When each singleton has its own unique color, we mean that each singleton in the finite collection is unique, or the collection has the minimum symmetry-degree.

Multiplication can be operated only among identical singletons, where addition is operated among unique singletons.

Each natural number is used as some given quantity, where in this given quantity we can order several different sets, that have the same quantity of singletons, but they are different by their symmetrical degrees.

In more formal way, within the same quantity we can define all possible degrees, which existing between a multiset and a "normal" set, where the complete multiset and the complete "normal" set are included too.

If we give an example of transformations between multisetes and "normal" sets, we can see that the internal structure of n+1 > 1 ordered forms, constructed by using all previous n >= 1 forms:

[b]1[/b]
(+1) = {x}

[b]2[/b]
(1*2)    = {x,x}
((+1)+1) = {{x},x}

[b]3[/b]
(1*3)        = {x,x,x}
((1*2)+1)    = {{x,x},x}
(((+1)+1)+1) = {{{x},x},x}

[b]4[/b]
(1*4)               = {x,x,x,x} <------------- Maximum symmetry-degree, 
((1*2)+1*2)         = {{x,x},x,x}              Minimum information's 
(((+1)+1)+1*2)      = {{{x},x},x,x}            clarity-degree
((1*2)+(1*2))       = {{x,x},{x,x}}            (no uniqueness) 
(((+1)+1)+(1*2))    = {{{x},x},{x,x}}
(((+1)+1)+((+1)+1)) = {{{x},x},{{x},x}}
((1*3)+1)           = {{x,x,x},x}
(((1*2)+1)+1)       = {{{x,x},x},x}
((((+1)+1)+1)+1)    = {{{{x},x},x},x} <------ Minimum symmetry-degree,
                                              Maximum information's  
                                              clarity-degree                                            
                                              (uniqueness)
[b]5[/b]
...

Can someone give me an address of some mathematical brach that researches these kind of relations between multisets and "normal" sets?

Thank you,

Organic

 

[Organic]

Since no one knows what mathmatical branch deals with connections between multisets and "normal" sets as I show in the previous post, I think maybe there is here a new Mathematical brach.