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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81
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2answers
18k views

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 ...
45
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8answers
45k views

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 ...
40
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7answers
23k views

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 ...
39
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0answers
1k views

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 minumum, to complete my task? As ...
37
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5answers
28k views

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 ...
32
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2answers
6k views

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.
31
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4answers
23k views

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.
30
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6answers
2k views

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 \...
26
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2answers
1k views

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 ...
25
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2answers
2k views

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 ...
24
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1answer
1k views

Are there any “crazy” totally ordered rings without infinities/infinitesimals?

I was seeing this post and its nice answer. I asked myself if I could have found a more "crazy" example. I was not very succesful. So I formalized what I was trying to achieve: Question: Let $(R,+,\...
23
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2answers
806 views

Are irrational numbers order-isomorphic to real transcendental numbers?

I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers? I know that the order type of ...
22
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6answers
5k views

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 ...
22
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1answer
2k views

What do linearly ordered abelian groups look like?

Recently a post on MathOverflow connected a theorem about ordered abelian groups to the axiom of choice, and it made me want to try and play with these objects a bit. But it appears to me that I don'...
19
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4answers
364 views

Is there a name for this kind of “betweenness structure”?

A homeomorphism $\mathbb R\to\mathbb R$ is almost the same thing as an order isomorphism, except that a homeophorphism can also be an order anti-isomorphism. I'm wondering whether there is a natural ...
19
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2answers
2k views

Order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$

Do you have a simple example of order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$? On $\mathbb Q$, I use the usual order and on $\mathbb Q\times\mathbb Q$, I define the ...
18
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6answers
11k views

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 ...
18
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5answers
5k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
18
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1answer
1k views

bijective homomorphisms between non isomorphic posets, example , explanation needed

Could someone please give a super simple example of a "bijective homomorphism between a non isomorphic poset"? I don't understand the sentence marked by green in the picture taken from Awodey's book.
18
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2answers
207 views

strictly increasing function from reals to reals which is never an algebraic number

Let $f:\Bbb R\rightarrow\Bbb R$ have the properties $\forall x,y\in\Bbb R,\space x<y\implies f(x)<f(y)$ and $\forall x\in\Bbb R,\space f(x)\notin\Bbb A$ where $\Bbb A$ is the set of algebraic ...
17
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5answers
4k views

Isn't every totally ordered set well-ordered?

A well order is a total order on a set $S$ with the property that every non-empty subset of $S$ has a least element. But surely it follows from the definition of a total order that any non-empty ...
17
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2answers
432 views

Is $(\pmb{1} + \pmb{\eta})\cdot\pmb{\omega_1} = \pmb{1} + \pmb{\eta}\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
17
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2answers
577 views

Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
17
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3answers
2k views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
16
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5answers
2k views

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 ...
15
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8answers
4k views

Can I always order a countable set of numbers

Suppose I have a countable subset of the real numbers $A$. Then since it is countable there is a function $f: \mathbb N \to A$ which is bijective. This function is of course a sequence, so we can ...
15
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1answer
485 views

Is this a characterization of well-orders?

While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
15
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2answers
405 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
14
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3answers
4k views

Sole minimal element: Why not also the minimum?

A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$. The minimum (at most one) of a partially ordered set $S$ is an ...
14
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4answers
4k views

Formal proof for A subset of the real numbers, well ordered with the normal order of $\mathbb R$, is at most $\aleph_0$

I tried to write a formal proof for the theorem: $A$ subset of $\mathbb R$ well ordered by the normal order $\implies A$ is at most of cardinality $\aleph_0$. Any suggestions? Thanks.
14
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1answer
6k views

What is meaning of strict weak ordering in layman's term? [closed]

I gone through many pages using Google, but not understand exact meaning of Strict-weak Ordering term. I have this requirement while sorting strings.
14
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2answers
584 views

An order type $\tau$ equal to its power $\tau^n, n>2$

In this question we are concerned only with linear (aka total) order types. By a cardinality of an order type we understand a cardinality of an instance of this type, which obviously does not depend ...
14
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1answer
221 views

Representation theorem for Complete Atomic Heyting Algebras

It is well known that within classical logic one can characterize complete atomic boolean algebras as powersets. Is it possible to provide any characterization/representation theorem for complete ...
13
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5answers
16k views

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 ...
13
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4answers
13k views

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.
13
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4answers
527 views

A “Cantor-Schroder-Bernstein” theorem for partially-ordered-sets

If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?
13
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2answers
704 views

Any good decomposition theorems for total orders?

I'm learning about set theory and partial orders, well orders, total orders, etc. I'm curious to know of any good decomposition theorems that apply to all (or very general types of) total orders. ...
13
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1answer
668 views

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 ...
13
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2answers
365 views

Ordinal interpretation of Friedman's $n$?

I heard that Kruskal's tree theorem can be turned into a finite form that creates an extremely fast growing function because ordinals could be encoded into trees. On this wiki page it mentions that ...
12
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3answers
3k views

Show that it is impossible to list the rational numbers in increasing order

This is problem #6 from Section 1.2 of Ash's Real Variables With Basic Metric Space Topology. I am asked to show that it is impossible to list the rational numbers in increasing order. While I ...
12
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3answers
1k views

Is there an example of an ordered ring that is not isomorphic to any subring of the real numbers?

Somebody told me it's possible to show the existence of rings non-isomorphic to any subring of the real numbers using model theory, but they couldn't get an example.
12
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1answer
605 views

$\mathbb{R}$ and $\mathbb{R}\setminus \{0\}$ are not isomorphic as linear orders.

Informal argument So, we know that $\mathbb{R}$ with its usual topology is connected, and of course $\mathbb{R}\setminus\{0\}$ is not. Any $``$supposed" isomorphism should grant us a homeomorphism ...
12
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2answers
1k views

The p-adic numbers as an ordered group

So I understand that there is no order on the field of p-adic numbers $\mathbb{Q_p}$ that makes it into an ordered field (i.e.) compatible with both addition and multiplication. Now, from the ...
12
votes
2answers
414 views

Is every suborder of $\mathbb{R}$ homeomorphic to some subspace of $\mathbb{R}$?

Let $X \subset \mathbb{R}$ and give $X$ its order topology. (When) is it true that $X$ is homeomorphic to some subspace of $Y \subset \mathbb{R}$? Example: Let $X = [0,1) \cup \{2\} \cup (3,4]$, then ...
12
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1answer
198 views

Is there a nice well-behaved order isomorphism between the real algebraic numbers and the rationals?

Via Cantor's back-and-forth method we know that the linearly ordered set of all rational numbers and the linearly ordered set of all real algebraic numbers are isomorphic. But from the point of view ...
11
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6answers
3k views

Is there a real number which comes just after a particular real number?

This is like when we say that the integer which comes just after $2$ is $3$. Is there a real number which comes just after a particular real number? For example: Is there a number which comes just ...
11
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5answers
1k views

Must a sequence be well-founded?

Must a sequence be well-founded? Is $\Bbb Z=(\ldots-1,0,1,2,\ldots)$ a sequence? Conventionally we think of $\Bbb N=(0,1,2,3,\ldots)$ as a sequence, but what about if it has no starting value? ...
11
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4answers
9k views

what is total order - explanation please

sorry for the dumbest question ever, but i want to understand total order in an intuitive way, this is the defition of total order: i) If $a ≤ b$ and $b ≤ a$ then $a = b$ (antisymmetry); ii) If $a ≤...
11
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4answers
433 views

Given an infinite poset of a certain cardinality, does it contains always a chain or antichain of the same cardinality?

It is known that given an infinite poset, it always contains an infinite chain or antichain; moreover, there is a constructive proof that we can find a continuous chain in $P(\mathbb{N})$; so, in ...
11
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1answer
2k views

Totally ordering the power set of a well ordered set.

Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$. How does a body actually use the well ordering of $...