# Tagged Questions

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.

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### Is every totally ordered finite dimensional vector space a lexicographic order for some basis?

Let's say we have a finite-dimensional vector space $V$ over a totally ordered field $\mathbb{K}$. Is every choice of totally ordered vector space structure (i.e compatible with the addition and ...
5answers
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### Examples of Galois connections?

On TWF week 201, J. Baez explains the basics of Galois theory, and say at the end : But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If ...
2answers
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### Q: Trichotomy of order of the real numbers

I am currently reading Terence Tao's "Analysis I" and while progressing through the book, the reader is repeatedly asked to prove trichotomy properties of order for the natural numbers $\mathbb{N}$, ...
3answers
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### Finding all partial order relations on a set

Suppose I have a set $A$ such that $A$ = $\{1, 2, 3, 4, 5\}$ (or $A$ = $\{1, 2, 3, 4\}$ or $A$ = $\{1, 2, 3\}$ or any other finite small set). How can I find the total number of partial order ...
1answer
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### Well ordering of the subsets of a given set

For a given set, does there always exists a well-ordering of the set of all its subsets which is stronger than the usual ordering (that is set-theoretic inclusion) of the sets of the subsets of the ...
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1answer
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### Is it true that no linear order can have CCC?

I am reading about countable chain condition on Wikipedia https://en.wikipedia.org/wiki/Countable_chain_condition I want to quickly verify that if $(X, \leq)$ is a linear order, then it cannot ...
2answers
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### Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
2answers
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### Find an example such that $X$ with the lexicographic order is not well-ordered.

Let $\{A_n\}_{n\in\Bbb N}$ be a collection of well-ordered sets. $X$ is defined by $X=\prod_{n\in\Bbb N}A_n$. Find an example such that $X$ with the lexicographic order is not well-ordered. I know ...
1answer
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### Are $(\mathbb R\times \mathbb Q, \le_h)$ and $(\mathbb Q\times \mathbb R, \le_h)$ isomorphic? [closed]

Are $(\mathbb R\times \mathbb Q,\leq_h)$ and $(\mathbb Q\times \mathbb R, \leq_h)$ isomorphic? when "$\leq_h$" is the right lexicographic order? Thanks a lot!
3answers
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### What does it mean to say “a divides b”

I am not a number theorist and I am learning about relations. I encountered a relation that says $a \leq b$ if $a$ divides $b$ Can someone clarify what it means to a number to divide another ...
1answer
651 views

### Are there only 2 clopen sets on real plane?

How can I prove that the only open and closed sets on the real plane are empty set and real plane itself? Preferably by using order theory. Thanks.
3answers
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### Preference relations that are order-separable

This question concerns dense order relations and separability of orders. This question has changed radically since the first version. Hence the first two answers below now look strange because they ...
3answers
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### A well-order on a uncountable set

I can't find an example of a well-order on an uncountable set. Is possible to prove that exists with the Axiom of Choice? How can I give a pratical construction? I try to define a well-order on ...
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2answers
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### How many ways can all six numbers in the set $\{4, 3, 2, 12, 1, 6\}$ be ordered

Is there an easy way to solve the problem? How many ways can all six numbers in the set $S = \{4, 3, 2, 12, 1, 6\}$ be ordered so that $a$ comes before $b$ whenever $a$ is a divisor of $b$? By ...