# Intersections of total pre-orders that preserve totality.

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.

• Yes, there are many examples. There are also many examples of functions that are continuous and many examples of sets that are finite. Commented Jun 9, 2017 at 20:05
• Likewise there are many cases where the specified B cannot be found. So what are you asking? Commented Jun 10, 2017 at 2:58
• Is there any difference between a total preorder and a total order? Additionally, if a preorder can be represented by collection of subsets, would it not have to be at least a partial order? Commented Jun 10, 2017 at 3:03
• @WilliamElliot yes I know that there are cases when such a B can't be found. I also say that. I speak of the cases when it can be found. I was asking for this because I'm interested in those cases when we can find such a subset---preferably maximal ones. I need this, because I'm defining an operation in ordering semantics to define a condition for a kind of relevant inference. In my case there merely existing cases where such subsets can be found is sufficient. Commented Jun 10, 2017 at 7:38
• @WilliamElliot think of a total preorder on some set A as a total order on subsets of A via the subset relation. Think of those subsets of A as equivalence classes of some sort. A total order is just a total preorder where the equivalence relation ~ is the identity relation. The general notion of preorder doen't insist on ~ as being the identity relation. A preorder would be a partial order if in addition to reflexivity and transitivity satisfied antisymmetry. I'm working with constructions similar to this one en.wikipedia.org/wiki/Preorder#Constructions Commented Jun 10, 2017 at 8:16