Julio - maybe someone already said what I'm about to say, or maybe you already know it, and it's probably not your main concern, but anyway.. in [comment #88](https://forum.azimuthproject.org/discussion/comment/18708/#Comment_18708) you wrote:

> I managed to understand the adoption of "≤" in \\( \mathcal{X}(x,y)\otimes\mathcal{X}(y,z)\leq\mathcal{X}(x,z) \\) via this angle, but am still not sure I fully grasp its necessity in \\(I\leq\mathcal{X}(x,x)\\) - I (think I) see that \\(\mathcal{X}(x,x)\\) need not be exactly \\(I\\), but why must it be _at least_ that much? What would happen if \\(I>\mathcal{X}(x,x)\\)?

This makes me feel nervous. Why? Because these are not the only two alternatives! In a poset,

not \\(I\leq\mathcal{X}(x,x)\\)


does not imply

\\(I\gt \mathcal{X}(x,x)\\) .


Only in a poset that is [totally ordered](https://en.wikipedia.org/wiki/Total_order) does the former imply the latter. In fact this is the definition of a totally ordered set!

We're interested in lots of posets that aren't totally ordered, like the poset \\(P(X)\\) consisting of all subsets of \\(X\\) . If \\(S\\) and \\(T\\) are subsets of \\(X\\),

not \\(S \subseteq T\\)


does not imply

\\( S \supset T \\) .