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 algebraic structures.

Order theory is a branch of mathematics which investigates the properties and structure of partial orders and quasi-orders (or preorders).

These sorts of orders appear naturally in the mathematical universe; such as the $\subseteq$ relation, or $\leq$ on the integers. It follows that these orders (and quasi-orders) appear in our lives as well, e.g. "which item is more expensive?"

Some mathematical concepts related to order theory are:

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Is there a known well ordering of the reals?

So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: Has anyone constructed a well ordering on…
Seamus
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How to find a total order with constrained comparisons

There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task? As a partial answer, I know that is possible to…
Jack D'Aurizio
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What does "curly (curved) less than" sign $\succcurlyeq$ mean?

I am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq) $$f(x^*) \succcurlyeq \alpha$$ and curved less than or equal to (\preccurlyeq) $$f(x^*) \preccurlyeq \alpha$$ Can someone explain…
Dinesh K.
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Example of Partial Order that's not a Total Order and why?

I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two. An explanation of why the example is a partial order but not a total order would also be greatly…
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difference between maximal element and greatest element

I know that it's very elementary question but I still don't fully understand difference between maximal element and greatest element. If it's possible, please explain to me this difference with some examples etc. I tried to explain this difference…
exTyn
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Every partial order can be extended to a linear ordering

How do I show that every partial order can be extended to a linear ordering? I think that I manage to prove that claim for finite set, how can I prove it for infinite set? Thank you.
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A finite set always has a maximum and a minimum.

I am pretty confident that this statement is true. However, I am not sure how to prove it. Any hints/ideas/answers would be appreciated.
CoffeeIsLife
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Infimum and supremum of the empty set

Let $E$ be an empty set. Then, $\sup(E) = -\infty$ and $\inf(E)=+\infty$. I thought it is only meaningful to talk about $\inf(E)$ and $\sup(E)$ if $E$ is non-empty and bounded? Thank you.
Alexy Vincenzo
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Embedding ordinals in $\mathbb{Q}$

All countable ordinals are embeddable in $\mathbb{Q}$. For "small" countable ordinals, it is simple to do this explicitly. $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1\}$. $\omega*2$ can be done…
Desiato
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What is the difference between max and sup?

I am studying KS (Kolmogorov-Sinai) entropy of order q and it can be defined as $$ h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right) $$ Why is it defined as supremum over all possible partitions P and not maximum? When do…
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Does Birkhoff - von Neumann imply any of the fundamental theorems in combinatorics?

I recently had the occasion to think about Hall's Marriage Theorem for the first time since my undergraduate combinatorics class more than a decade ago. Reading the wikipedia article linked above, I was interested to see that it is regarded as…
Pete L. Clark
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Is the set of real numbers the largest possible totally ordered set?

Because I find any totally ordered set can be "lined up" in a straight line, I'm guessing that the set of all the real numbers is the biggest totally ordered set possible. In the sense that any other totally ordered set is isomorphic…
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What is the order-type of the set of natural numbers, when written in alphabetical order?

We are all familiar with the standard nomenclature for the smallish natural numbers, such as one, two, three, ..., one hundred, one hundred one, ..., fifteen thousand two hundred forty-nine. I have in mind the simple American number…
JDH
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Why do people accept the axiom of choice given the well ordering principle?

We know without any doubt that the axiom of choice implies (in fact is equivalent to) the well ordering principle. The well ordering principle can't be true! If we take the open interval $(0,1)$ for example, there can't be a least (or most) element.…
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Simplest Example of a Poset that is not a Lattice

A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a partially ordered set can fail to be a lattice.…
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