Terminology: canonical member of equivalence class
An equivalence relation $R$ partitions a set $X$ into equivalence classes. It is common to denote the equivalence class of an element $x \in X$ as $[x] = \{u \in X \mid u R x\}$. However, this leads to a given equivalence class having multiple names: one for each element of the class. Is there a convention identifying a canonical element/name of a class, or a name for this canonical label?
As an example, let’s use the congruence relation modulo 3 on the natural numbers $\mathbb N^+$. There are three equivalence classes. One of them can reasonably be called $[1]$, $[4]$, $[136]$, $[850000123]$, or by infinitely many other names.
To make it clear that we are always referring to the same thing, we may choose to always call this class $[1]$, selecting 1 as a privileged representative of the class. (I chose it because it is the minimum element, but the criterion seems unimportant, so long as there is a criterion.)
Is there a name, either for a privileged representative $k$ of a given class, or for a canonical label $[k]$?
$\endgroup$ 21 Answer
$\begingroup$You touch a nontrivial question since the existence of such a selector function is the axiom of choice. Hilbert's epsilon calculus and Bourbaki's tau-square notation are two formal approaches for this question.
That being said, there is often in practice a natural choice. For instance, for the case of the congruence modulo $k$ on $\Bbb N$, you can choose the rest of the Euclidean division of $n$ by $k$ to represent the equivalence class.
$\endgroup$ 1More in general
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