A Sketch of a Formalisation of Self Similarity

Introduction

I'd like to present a useful formalism for describing when a set^[1] is "self-similar".

Isomorphism Under Equivalence Relations

Given arbitrary sets $X,Y$, an "equivalence-isomorphism" is a tuple $(f:X\to Y,f^{-1}:Y\to X,{\sim }_R\subset X\cup Y\times X\cup Y)$, such that:

Where:  

• $f$ is a bijection from $X$ to $Y$
• $f^{-1}$ is the inverse of $f$
• ${\sim }_R$ is an equivalence relation on the union of $X$ and $Y$.  

For a given equivalence relation ${\sim }_R$, if there exist functions such that an equivalence-isomorphism can be constructed, then we say that the two sets are "isomorphic under ${\sim }_R$".

The concept of "isomorphism under an equivalence relation" is meant to give us a more powerful mechanism for describing similarity/resemblance between two sets than ordinary isomorphisms afford^[2].

Similarity

Two sets are "similar" if they are isomorphic to each other under a suitable equivalence relation^[3].

Self-Similarity

A set is "self-similar" if it's "similar" to a proper subset of itself.

Closing Remarks

This is deliberately quite bare, but I think it's nonetheless comprehensive enough (any notion of similarity we desire can be encapsulated in our choice of equivalence relation) and unambiguous given an explicit specification of the relevant equivalence relation. 
 

1. ^{^}

   I stuck to sets because I don't know any other mathematical abstractions well enough to play around with them in interesting ways.

2. ^{^}

   Ordinarily, two sets are isomorphic to each other if a bijection exists between them (they have the same cardinality). This may be too liberal/permissive for "similarity".

   By choosing a sufficiently restrictive equivalence relation (e.g., equality), we can be as strict as we wish.

3. ^{^}

   Whatever notion of similarity we desire is encapsulated in our choice of equivalence relation.