This is a subject too modest to be considered good mathematics. It is a sort of attempt to lay out when various structures like lattices, and orderings are useful.

I think that there is no more to a preorder than a partial order among equivalence classes. Further a preorder is merely a canonical form of a transitive relation with the convention that a ≤ a. This may make it quicker to decide if a preorder is useful in some particular application. Such observations help, I think, in the task of applying such ideas to real tasks such as programming. They are known by any good mathematician, but amateurs could use some guidance. The math is presented here, the pragmatics are not. Finch counts these things.

There are sever similar ideas: Are there useful unique latices in which a partial is included? Just as in practice we use a many real numbers but find it useful to reason about the much more numerous set of all the reals; is it similarly useful to reason about these containing lattices?

Another sort of question is “How profound is the topologist’s decision to decree (via axioms) that the null set is open?” To go against that rule would make some theorems a bit longer and others a bit shorter. I would expect that it is efficient and merely efficient (of theorem and proof space) that the null set must be open.

Many years ago I noticed that the axioms for first order predicate calculus did not allow the null model. I found that it was easy to modify the axioms to admit the null model. It didn’t seem to make much difference in a few metatheorems with which I was then familiar. I think that I proved, as a metatheorem, that there could be no profound consequences to such a change. It also seems unlikely that slightly more permissive axioms would have led to insights otherwise missed.

(X ∧ M) ∨ (Y ∧ ~M) = (((X ⊕ Y) ∧ M) ⊕ Y).
A similar issue has arisen expressed in graph theory which is hardly different from relation theory. Given a directed graph T, find a way to represent its transitive closure so as to enable economical access. Sometimes the following works well. Find the strongly connected components, the set of this is D. D is a DAG. Form a computational map from T to D. Represent D by whatever algorithm you like.

There are two sorts of tension here:

  1. Between graph theory and relation theory
  2. Between computer science and math.
It would be good to elucidate these issues.