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Suppose there is a set of objects on which we can define an equivalence relation. Under some transformations of the space on which the objects are defined, these objects may change their equivalence class. As a trivial example consider vectors on a $2$ dimensional euclidean space and the equivalence relation defined by $a \sim b$ if their $x$-components have the same sign or they are both null. In this example there are $3$ equivalence classes, but under a change of coordinates one vector may transition from one equivalence class to another. In this case I suppose the equivalence relation should be defined on each particular coordinate system.

I want to find out whether there may be any interesting properties in such cases. Could anyone provide some insight or literature on this subject if there is any? I could not find anything relevant on this topic.

Sarvesh Ravichandran Iyer
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Damian
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    I'll stick to the example you provided. Suppose $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map that for every equivalence class $[x] \subset \mathbb{R}^2$ for some $x \in \mathbb{R}^2$ maps any element of $[x]$ to the same equivalence class. Then $f$ passes down to a well-defined map on the quotient set that acts on the cosets. If $f$ is the reflection along the $y$-axis for instance, $f$ permutes the signed equivalence classes and preserves the equivalence class corresponding to the $y$-axis.

    I do not know what your background is, but one interesting application would be Markov Partitions.

     – Lukic May 18 '22 at 15:36
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    In the example, the equivalence relation is necessarily defined with reference to a coordinate system (otherwise what does "$x$-component" mean?), so you get a different relation when using a different coordinate system. On the other hand, if you define a relation using only the language of abstract vector spaces (e.g. $x\sim y$ iff $\exists \alpha\ne0,y=\alpha x$) then it will indeed be preserved by any vector space isomorphism. – Karl May 18 '22 at 15:48

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