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.
<p>I think that there is no more to a <a href=preorder.html>preorder</a>
than a <a href=po.html>partial order</a> among equivalence classes.
Further a  preorder is merely a canonical form of a
<a href=trans.html>transitive relation</a> with the <i>convention</i> that a &#8804; 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 <a href=http://www2.units.it/etica/1999_2/floridi/node9.html>here</a>, the pragmatics are not.
Finch <a href=http://www.people.fas.harvard.edu/~sfinch/csolve/posets.pdf>counts these things</a>.
<p>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?
<p>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.
<p>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.
<hr>
(X ∧ M) ∨ (Y ∧ ~M) = (((X ⊕ Y) ∧ M) ⊕ Y).
<hr>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.
<p>There are two sorts of tension here:<ol>
<li>Between graph theory and relation theory
<li>Between computer science and math.</ol>
It would be good to elucidate these issues.