I am barely new to order theory and this motivates if the question is trivial.
I understood the definitions of preorder, partially and totally ordered sets and well ordered sets.
In particular there is a nice hierarchy among the last three and I think that for a newcomer these are the most natural enviroments to think of, as one is used to the real line and power sets. Moreover these three capture the "natural" idea of order I may think of, at different levels.
On the other hand directed set are less intuitive to me. Clearly do not fit into the hierarchy since they do not libk with partially ordered sets, but totally ordered sets are directed, even if the converse does not hold. Hence I ask: in which sense directed sets generalize totally ordered sets? Which is the intuition behind them? What aspects of order they capture and what motivates the name directed?
Best Answer
My intuition behind directed sets is that they give information about some bigger object, but every element in the directed set only gives a small amount of information. Then combining a finite amount of these elements still gives a small amount of information.
Let me illustrate that with an example: consider some infinite set $X$. Then let $\mathcal P_{fin}(X)$ denote the set of all its finite subsets. We can naturally order $\mathcal P_{fin}(X)$ by inclusion, so it becomes a directed poset. Now how does this fit the intuition I just described? Well, every element in $\mathcal P_{fin}(X)$ is a finite subset of $X$, so it gives us some information about what is in $X$, but only a small amount. The fact that $\mathcal P_{fin}(X)$ is directed says that for any $U_1, \ldots, U_n \in \mathcal P_{fin}(X)$ we can find some upper bound $V$ in $\mathcal P_{fin}(X)$. This $V$ will still be finite, so combining a finite amount of 'small' pieces of information we end up with a 'small' piece of information.
For different kinds of mathematical objects, the same kind of intuition holds. For example, for any kind of algebraic object (e.g. vector spaces, groups, rings, etc.) or even models of a first-order theory can be decomposed in 'smaller' pieces in this way. For example for an infinite dimensional vector space, we could look at all the subspaces spanned by finitely many vectors.