[red-herring comment](https://forum.azimuthproject.org/discussion/comment/18489/#Comment_18489)
The poset vs. preordered set vs. partially-order set, discussion that happened earlier seems to be a case of the mathematical red herring principle almost happening.

When we start with a set, let's call it \\( \mathbb{X} \\) and some relation \\( \oplus \\) we do this...
$$ \mathcal{X} := ( \mathbb{X}, \oplus ) $$
...and then apply some rules governing the behavior of \\( \oplus \\) over \\( \mathbb{X} \\).
What do we call what was just done and what was formed namely \\( \mathcal{X} \\)?

One more question.
Is this a case where all \\( \mathcal{V}\text{-categories} \\) are \\( \text{categories} \\) but not all \\( \text{categories} \\) are \\( \mathcal{V}\text{-categories} \\)?
Surely there will be some relationship between them [maybe I should just be patient and all will be revealed].