# Questions tagged [order-theory]

Order theory deals with properties of orders, usually partial orders or quasi orders but not only those. Questions about properties of orders, general or particular, may fit into this category, as well as questions about properties of subsets and elements of an ordered set. Order theory is not about the order of a group nor the order of an element of a group or other alegbraic structures.

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### Which of the following are Hasse Diagrams?

My assumptions are, the first and the last diagrams are Hasse. Can someone please explain which one is a Hasse and why?
1answer
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### Strict WQOs and Strict WPOs

The Wikipedia article on WQOs does not mention a strict version. I came across a particular relation, which I could only describe as a strict WQO, but I am wondering if my reasoning is correct and if ...
2answers
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1answer
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### If a poset is $\sigma$-centered, so what can we say about $P\times P$

We say that a poset $P$ is $\sigma$-centered if it can be partitioned into countably-many pieces so that each piece is finite-wise compatible. i.e. it is $\sigma$-centered if there exists a partition ...
0answers
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### Filters on preordered sets vs filters on partially ordered sets

What is the advantage on defining filters on partially ordered sets versus defining them on preordered sets? Almost everywhere where I have seen filters they have been used in the setting of partially ...
1answer
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### A partial order on $\mathbb{N}$ that is not total

In the text Modern Real Analysis by Ziemer there is a question that asks to use the "natural partial order" on $\mathcal{P}(\{1,2,3\})$ to obtain a partial order on $\mathbb{N}$. I have scratched my ...
1answer
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1answer
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### Are there subsets of $\mathbb{Q}$ with a finite number of upper-bounds?

Consider the poset $(\mathbb{Q}, \leq)$. Is there a subset of $\mathbb{Q}$ with a finite number of elements that are upper-bounds? I tried to prove this as follows: Suppose $K \subset \mathbb{Q}$, ...
2answers
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### example of non order preserving bijection

I read that "the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both ...
4answers
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