Questions tagged [order-theory]

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43 votes
4 answers
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A principle of mathematical induction for partially ordered sets with infima?

Recently I learned that there is a useful analogue of mathematical induction over $\mathbb{R}$ (more precisely, over intervals of the form $[a,\infty)$ or $[a,b]$). It turns out that this is an old ...
Pete L. Clark's user avatar
41 votes
3 answers
2k views

What is the minimal size of a partial order that is universal for all partial orders of size n?

A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds order-preservingly into $\mathbb{B}$. For example, every partial order $\langle\...
Joel David Hamkins's user avatar
35 votes
3 answers
2k views

How much choice is needed to show that formally real fields can be ordered?

Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...
Pete L. Clark's user avatar
34 votes
11 answers
3k views

Open questions about posets

Partially ordered sets (posets) are important objects in combinatorics (with basic connections to extremal combinatorics and to algebraic combinatorics) and also in other areas of mathematics. They ...
34 votes
5 answers
1k views

Does the exact pair phenomenon for partial orders occur in your area of mathematics?

Suppose that I have a partial order P and an increasing sequence $a_0< a_1<a_2<\cdots$ of elements of $P$. A pair of elements (b,c) from P is said to be an exact pair for this sequence, if ...
Joel David Hamkins's user avatar
33 votes
7 answers
4k views

What's a non-abelian totally ordered group?

Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with a well-ordering) ...
Andrew Critch's user avatar
32 votes
9 answers
5k views

How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ...
Joel David Hamkins's user avatar
31 votes
3 answers
2k views

Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
Mostafa Mirabi's user avatar
28 votes
1 answer
6k views

What is the cofinality of the co-infinite subsets of ${\bf N}$?

Let ${\mathcal A}$ be the family of subsets $A$ of the natural numbers ${\mathbf N}$ which are co-infinite (i.e., their complement is infinite). We partially order this family by set inclusion. A ...
Terry Tao's user avatar
  • 108k
26 votes
3 answers
2k views

When does a graph underlie the Hasse diagram of a poset?

For any finite poset $P=(X,\leq)$ there is a graph $G$ underlying its Hasse diagram $H=(X,\lessdot)$, so that $V(G)=X$ and $E(G)=\{\{u,v\}:u\lessdot v\}$. With that said, is it possible to ...
Ethan Splaver's user avatar
26 votes
3 answers
2k views

When does a Galois connection induce a topology?

Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps $\Phi: X \rightarrow Y, \ \Psi: Y \...
Pete L. Clark's user avatar
25 votes
1 answer
1k views

Expected height of a poset?

I am interested in any known results/empirical studies done on the average height of a poset with $N$ elements. Obviously this would depend on how that poset relation was randomly defined, however, at ...
AspiringMat's user avatar
25 votes
0 answers
536 views

A conjecture about inclusion–exclusion

$\newcommand\calF{\mathcal{F}} \def\cupdot {\stackrel{\bullet}{\cup}} \def\minusdot {\stackrel{\bullet}{\setminus}}$This post presents a conjecture that we have with some colleagues. It is about ...
M.Monet's user avatar
  • 351
24 votes
2 answers
660 views

Order type of the smallest set containing the identity function and closed under exponentiation

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$...
TauMu's user avatar
  • 872
23 votes
3 answers
2k views

Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?

Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?
Marty Colos's user avatar
23 votes
2 answers
1k views

Which are the rigid suborders of the real line?

Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...
Joel David Hamkins's user avatar
22 votes
1 answer
2k views

Why do we need "canonical" well orders?

(I asked this question on Math.SE earlier but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-...
Vivaan Daga's user avatar
21 votes
2 answers
1k views

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

(This is a re-post of my old unanswered question from Math.SE) For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types. Recall that: $...
Vladimir Reshetnikov's user avatar
19 votes
4 answers
1k views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
Martin Brandenburg's user avatar
19 votes
1 answer
1k views

Is the theory of a partial order bi-interpretable with the theory of a pre-order?

A partial order relation $\leq$ on a set $A$ is a binary relation that is reflexive, transitive, and antisymmetric. A preorder relation $\unlhd$ (also sometimes known as a quasi order or pseudo order) ...
Joel David Hamkins's user avatar
19 votes
1 answer
457 views

Large Borel antichains in the Cantor cube?

Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
Taras Banakh's user avatar
  • 40.8k
19 votes
0 answers
765 views

A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions $$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$ Suppose we say that $...
Dmitry V's user avatar
  • 433
18 votes
3 answers
759 views

What is the minimum size of a partial order containing all partial orders of size 5?

This earlier MO question asks to find the minimum size of a partial order that is universal for all partial orders of size $n$, i.e. any partial order of size $n$ embeds into it, preserving the order. ...
Caleb Stanford's user avatar
18 votes
1 answer
1k views

Suprema of directed sets

Let $(X, \le)$ be a partially ordered set. We call a subset $S \subseteq X$... ... a chain if each two elements in $S$ are comparable with respect to $\le$ (in other words, $S$ is linearly ordered ...
Jochen Glueck's user avatar
18 votes
3 answers
1k views

Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural ...
Tom LaGatta's user avatar
  • 8,372
18 votes
4 answers
1k views

Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, i....
François G. Dorais's user avatar
18 votes
2 answers
1k views

Can all $\aleph_2$-dense subsets of $\mathbb{R}$ be isomorphic?

Let $\kappa$ be an infinite cardinal. For a subset $A \subseteq \mathbb{R}$, we say that $A$ is $\kappa$-dense if $|A \cap (a, b)| = \kappa$ for every interval $(a, b)$. By Cantor, any two $\aleph_0$-...
Garrett Ervin's user avatar
18 votes
1 answer
1k views

Kuratowski closure-complement problem for other mathematical objects?

The original Kuratowski closure-complement problem asks: How many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given starting subset of a ...
Zev Chonoles's user avatar
  • 6,712
18 votes
1 answer
10k views

Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow. In non-Hausdorff topology it is standard to ...
polmath's user avatar
  • 321
17 votes
4 answers
3k views

Explicit ordering on set with larger cardinality than R

Is it possible to construct (without using Axoim of Choice) a totally ordered set S with cardinality larger than $\mathbb{R}$? Motivation: A total ordering is often called a “linear ordering”. I have ...
Sune Jakobsen's user avatar
17 votes
2 answers
332 views

Is it possible to reconstruct an order type from its initial segments?

Suppose $T$ is a totally ordered set without a maximal element, $\tau$ is the order type of $T$, $S$ is the set of order types of all proper initial segments (downward closed subsets) of $T$. Is it ...
Piotr Shatalin's user avatar
17 votes
1 answer
1k views

Has the exponentiation of ordinals a nice geometric model?

It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed ...
Taras Banakh's user avatar
  • 40.8k
17 votes
1 answer
1k views

How is this fixed point theorem related to the axiom of choice?

I'm hoping the answer to this is well-known. Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) \...
Tom Leinster's user avatar
  • 27.2k
16 votes
4 answers
1k views

Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way? Let $\mathcal{C}$ be a class of (...
Avshalom's user avatar
  • 2,121
16 votes
3 answers
10k views

Proving that a poset is a lattice

I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably ...
Martin Rubey's user avatar
  • 5,533
15 votes
5 answers
7k views

infinite permutations

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same ...
kakaz's user avatar
  • 1,596
15 votes
3 answers
714 views

Posets isomorphic to their endomorphism poset

Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise. Is there a poset with more than 1 point such that $P\cong \text{...
student9909's user avatar
14 votes
1 answer
1k views

Characterizing $\mathbf{R}$ as an ordered group

A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ...
coudy's user avatar
  • 18.5k
14 votes
1 answer
781 views

Does there exist an ordering-functor?

This sounds like a very silly question which should have have a negative answer but I don't see an argument. The precise question is this: Does there exist a covariant functor $ord$ from the category ...
Jochen Wengenroth's user avatar
14 votes
1 answer
312 views

Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$. ...
Dominic van der Zypen's user avatar
14 votes
5 answers
804 views

Birkhoff's representation theorem vs matroid-geometric lattice correspondence

This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says ...
Igor Makhlin's user avatar
  • 3,493
14 votes
1 answer
2k views

totally ordered chain in the powerset with big cardinality

Let $B$ be some set. The problem is to find a set $A\subset\mathcal{P}(B)$ of subsets of $B$ which is totally ordered by inclusion and such that there exists a bijection $A\leftrightarrow \mathcal{P}(...
Toink's user avatar
  • 622
14 votes
1 answer
349 views

Comparing sizes of sets of integers

Is there a total preorder $\lesssim$ on the power set of $\mathbb Z$ such that: $A<B$ if $A\subset B$ (proper subsets are smaller) $1+A\lesssim 1+B$ iff $A\lesssim B$ (where $1+C = \{1+c:c\in C\})...
Alexander Pruss's user avatar
14 votes
1 answer
1k views

Any further applications of Freudenthal's 1936 Spectral Theorem?

Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...
Elemer E Rosinger's user avatar
14 votes
1 answer
582 views

On certain order-automorphisms of the rationals

Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order. ...
THC's user avatar
  • 4,313
13 votes
2 answers
2k views

Ordinals that are not sets

The class of all ordinal numbers $\mathbf{Ord}$, aside being a proper class, can be thought of an ordinal number (of course it contains all ordinal numbers that are sets, not itself). Then one could ...
sdcvvc's user avatar
  • 928
13 votes
2 answers
678 views

How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?

How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$? How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$? I can see that results in both ...
Hanna K.'s user avatar
  • 233
13 votes
1 answer
262 views

Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin})$?

We endow ${\cal P}(\omega)$ with an equivalence relation by saying that $A\simeq_{\text{fin}} B$ iff the symmetric difference $A\Delta B$ is finite. The resulting set of equivalence classes is denoted ...
Dominic van der Zypen's user avatar
13 votes
1 answer
938 views

Does every feasible partial order relation on the natural numbers extend to a feasible linear order relation?

It is well known that every partial order on a set can be extended to a linear order on that set. That is, for every partial order $\lhd$ on a set $X$, there is a linear order $\prec$ on $X$ such that ...
Joel David Hamkins's user avatar
12 votes
4 answers
1k views

Universal order type

Every countable order type, such as the countable ordinals, $\mathbb Z$, etc. can be embedded in $\mathbb Q$, so it is universal for countable order types. Is there a universal space for all linear ...
mathahada's user avatar
  • 636

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