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I'm currently trying to express the following:

given a structured set of sets, of the form {1}, {1,{1,1}}, {{1,1},{1,1}}, or {1,{{1,1},1}} etc.

I want to be able to test whether two sets are reductions of each other, where reduction means, that:

{1,{1,1} is equal to {1}, or {1,1}, but not {{1,1},1}.

The logic behind reduction is that I can reduce a subset to a subset of smaller cardinality, or singleton set (so {1,1} is a reduction of {1,{1,1}}. The relation is transitive, so given {1} is a reduction of {1,1} then {1} is a reduction of {1,{1,1}}.

I think I have the right properties: given two ordered sets A and B, B is a reduction of A

1) iff A and B contain the same number of subsets, and there is a bijection between the subsets of A and the subsets of B, where the corresponding subset in B is itself a reduction of the corresponding subset in A. (basically what I'm trying to say is that if you have {1,2,3} and {a,b,c} then 1 is linked to a, 2 to b, etc. also works for subsets, where you have {{1,2},3,4} and {a,b,c} where in that case {1,2} is linked with a, 3 with b, and 4 with c)

2) iff there exists a set R such that B is a reduction of R, and R is a reduction of A.

Does this definition make sense? And if yes, I'm really struggling to provide formal notation for 1. I'm not very familiar with set notation.

Thanks!

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# Formulation of recursive subset equality

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