I juse met the defiition of dimension of proset and I found myself stuck on a question (actually I am having various troubles with it, so the one I am asking is only one part of the problem I found).
Assuming that $\Delta_X$ is the Diagonal Relation on an arbitrary set $X$:
1) what is $dim(X,\Delta_X)$?
2) what is $dim(X,X^2)$?
1) After quite a lot of thoughts, I think that $dim(X,\Delta_X)=2$. The intuition I had is that we have to imagine two chains, the first that moves from - let's say - $a_1$ to $a_n$, while the other chain moves in the opposite direction and the $a_i$ are the elements on which we define the diagonal relation. If we intersect the two chains, indeed we get the original diagonal relation.
2) About $dim(X,X^2)$, I think that it should be equal 1. My line of reasoning is the following: $dim(X,X^2)$ is already a complete preorder (actually it is - so to speak - the most complete), so there is nothing to extend and there is no intersection to work on.
Are my lines if reasoning ok?
Thanks in advance for any feedback.