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27
votes
3answers
813 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 ...
27
votes
2answers
640 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 ...
27
votes
4answers
2k views

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 ...
25
votes
9answers
2k 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. ...
23
votes
2answers
414 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 ...
22
votes
3answers
711 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 ...
21
votes
3answers
507 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?
20
votes
2answers
643 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 ...
19
votes
6answers
2k 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) ...
19
votes
4answers
897 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 ...
17
votes
2answers
662 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 ...
16
votes
2answers
254 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 ...
16
votes
2answers
738 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: ...
14
votes
3answers
3k 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 ...
14
votes
3answers
822 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 ${\leq}$, ...
12
votes
2answers
317 views

What is the least ordinal than cannot be embedded in $\mathbb{R}^\mathbb{R}$?

Let $\mathbb{R}^\mathbb{R}$ be the set of functions $\mathbb{R}\to\mathbb{R}$ patially ordered by eventual domination. Obviously, every ordinal below $\omega_1$ can be embedded in ...
12
votes
1answer
759 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 ...
12
votes
1answer
591 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 ...
11
votes
1answer
613 views

Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := ...
11
votes
2answers
424 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 ...
10
votes
5answers
1k 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 ...
10
votes
5answers
3k 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 ...
10
votes
2answers
323 views

Do operations generate well-ordered sets only?

I've read   @TauMu's question   about the set of functions   $\mathbb N\rightarrow\mathbb N$   generated from the identity map by repeatedly applying exponentiation of two already ...
10
votes
3answers
167 views

Maximal chains in a quasi-order of linear order types

Let $\mathcal{T}_\kappa$ be the set of all linear order types of cardinality $\kappa$. Let $\prec$ denote a binary relation on $\mathcal{T}_\kappa$ representing embeddability of order types (note that ...
10
votes
1answer
761 views

Discrete version of Nullstellensatz?

Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...
10
votes
1answer
572 views

Can all aleph_2-dense subsets of 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 ...
10
votes
1answer
222 views

Order dimension and weak poset partitions

The order dimension of a poset $(P,\leq)$ is the least number of linear extensions of $(P,\leq)$ such that the intersection of these extensions is $(P,\leq)$. The wikipedia entry provides some ...
10
votes
1answer
434 views

Partial word orders on groups

This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have ...
10
votes
1answer
769 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 ...
9
votes
3answers
212 views

Is the homomorphism poset directed if the codomain is directed?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. (We call a function $f:P\to Q$ order preserving if $x\leq y$ in $P$ ...
9
votes
2answers
302 views

Extending a partial order while preserving an automorphism

It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by ...
9
votes
1answer
314 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 ...
9
votes
0answers
196 views

Reference for sparseness of incomparability graphs implying sparseness of covering graphs

If a partial order on $n$ elements has $m$ incomparable pairs, then its covering graph (aka Hasse diagram aka transitive reduction, the graph of pairs of elements that are comparable but are not the ...
9
votes
1answer
423 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 ...
8
votes
4answers
797 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 ...
8
votes
6answers
2k views

Generalizations of Boolean posets/lattices

A Boolean lattice has a number of rather nice properties which give it a central role in many parts of combinatorics. For instance, it's a lattice, it can be augmented with a ring structure, it can ...
8
votes
4answers
754 views

Order types of positive reals

Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the ...
8
votes
5answers
927 views

Questions about ordering of reals and irrationals

Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4): 1) Is there a nondecreasing function from irrationals onto reals? 2) Is there a ...
8
votes
1answer
543 views

Which countable linear orders are $\aleph_0$-categorical?

The question is: Which countable linear orders are $\aleph_0$-categorical? I have a bit of progress on this: Define a discrete tuple to be a set of elements, ordered discretely, such that if $a$ and ...
8
votes
1answer
213 views

Does every countably infinite interval-finite partial order embed into the integers?

A partially ordered set $(S,\le)$ is called interval finite if the open intervals $(x,z):=\{y|x\le y\le z\}$ are finite for all choices of $x,z$ in $S$. An embedding $(S,\le)\rightarrow(S',\le')$ of ...
8
votes
1answer
383 views

Does this “flipping lexicographic” ordering have a standard name?

I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the flipping lexicographic ordering, for evident ...
8
votes
1answer
327 views

Decomposing posets into countably many chains

A conjecture of Galvin's is that the following is possible (which I take to mean that the consistency of the following can be proven relative to the consistency of something like ZFC, or ZFC plus some ...
8
votes
2answers
681 views

Is $Ded(\kappa)<Ded(\kappa)^\omega$ consistent?

Hello, I want to ask if anyone can tells us what is known (consistently) about $Ded(\kappa)$, $\kappa$ an infinite cardinal. Definition If there is a dense linear order w/o endpoints of size ...
8
votes
1answer
183 views

Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A ...
8
votes
0answers
263 views

Higher-order dimension in posets: a reference request

Let $P = (X, \le)$ be a partially-ordered set. Then the dimension of $P$ is the minimum number of total orders over $X$ whose intersection yields $P$. Alternately, the dimension of $P$ is the minimum ...
7
votes
3answers
315 views

Extracting countable chains from linear orders

There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{*}$ (by ...
7
votes
2answers
334 views

Ordinals and complexity classes

What is the least recursive ordinal $\alpha$ such that there is no algorithm in complexity class $\mathsf{P}$ which implements a well-ordering of $\mathbb{N}$ with order type $\alpha$? (where the size ...
7
votes
2answers
323 views

Which linearly ordered sets have the property that their completion is equipotent with their powerset?

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the ...
7
votes
1answer
345 views

A characterization of the poset of filters on a set

For the lattices of all subsets of a given set it is known an axiomatic characterization: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra. The ...
7
votes
1answer
270 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...