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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?

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

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  • $\begingroup$ I find your answer very interesting: do you know if this parallelism between infinite dimensional vector spaces/ finite dimensional subspaces on the one side and partially ordered sets/directed subsets can be developed somehow more in general? $\endgroup$ – Francesco Bilotta Apr 19 at 14:36
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    $\begingroup$ 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. $\endgroup$ – Mark Kamsma Apr 19 at 14:43
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    $\begingroup$ All of this is fundamental to the notion of a locally presentable category (or more generally, an accessible category). There the idea is that when we consider a certain kind of mathematical objects (e.g. vector spaces), then there is a set of 'small' ones that we can use to build all the other ones. This 'building' process is done by directed colimits, which basically means that we take a union of a directed diagram as I did in my answer. $\endgroup$ – Mark Kamsma Apr 19 at 14:47
  • $\begingroup$ I really thank you! I am just in my first years of math (I should get my BSc in months, i hope) so I never really studied category theory, although I heard a lot about it: btw do you know if there is some accessible book on category theory? I know the classic is Saunders-MacLane, but maybe it requires some more advanced background $\endgroup$ – Francesco Bilotta Apr 19 at 14:52
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    $\begingroup$ I think Category Theory by Steve Awodey is very accessible, if you have some mathematical background already (i.e. you know what groups are, topological spaces, etc. because these are often used as examples). It has the advantage that it spells out what most definitions actually mean, on the other hand it will not really train you in reading category-theoretic language. But if you are interested, you can learn that after you know what all the notions actually mean. $\endgroup$ – Mark Kamsma Apr 19 at 14:56

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