# Directed set and partially/totally ordered sets.

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?

• directed limits of groups/modules are easier to understand when the indexing set is directed. – Lord Shark the Unknown Apr 18 at 16:59

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.
• Certainly, what I just described is essentially a category-theoretic way of defining the 'size' of an object. I do not know how familiar you are with category theory, but the notion I am talking about is that of a "finitely presentable object" or "compact object" (this generalises easily to $\lambda$-presentable for any regular cardinal $\lambda$). See e.g. nLab, Wikipedia or this question. – Mark Kamsma Apr 19 at 14:43