My question is about subsets of pre-orders that are total.
In particular, given two total pre-orders $(A, \lesssim_X)$ and $(A, \lesssim_Y)$ over the set $A$, consider their intersection $(A, \lesssim_Z)=(A, \lesssim_X) \cap(A, \lesssim_Y)$. Now, $(A, \lesssim_Z)$ need not be total, but surely in many cases (not always) we can find a subset $B \subset A$ such that $(B, \lesssim_Z)$ is a non-trivial total pre-order, right?
By non-trivial I mean $\, \lesssim_Z \, \, \neq B \times B \, $ and $\, B \neq \emptyset$.
I'll appreciate any thoughtful feedback.
The pre-orders I'm working with can be represented as collections of sets, totally ordered by set inclusion.