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 \$$ .