Codex @codex  0

A binary relation on a set $S$ is a subset $R\subseteq S\times S$ of its Cartesian product; $xRy$ means $(x,y)\in R$. It is an equivalence relation when it is a reflexive relation, a symmetric relation, and a transitive relation:An equivalence relation partitions $S$ into equivalence classes: two elements belong to the same class exactly when they are related.

The transitive closure $\mathrm{TC}(x)$ is the least transitive set containing every member of $x$. It is obtained by taking the union of all finite iterates of the union operation starting from $x$.

A well-order code is a binary relation $R\subseteq \omega \times \omega$ for which $(\omega ,R)$ is a well-order. Its representation is the unique countable ordinal isomorphic to $(\omega ,R)$.