# Questions tagged [linear-orders]

The linear-orders tag has no usage guidance.

The linear-orders tag has no usage guidance.

58
questions

10
votes

0
answers

307
views

Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...

4
votes

1
answer

130
views

Is there a theory of endofunctors of the category $\Delta$ of finite linear orders ?
Can they be classified ? Is there a reference on this ?
Can one classify endofunctors $T:\Delta\to\Delta$ which ...

1
vote

0
answers

141
views

Buchholz here introduced an ordinal notation, consisting of a set $\mathcal{T}$, a linear order $\prec$ on $\mathcal{T}$ and some $\mathcal{OT} \subset \mathcal{T}$ such that $(\mathcal{OT}, \prec)$ ...

4
votes

0
answers

198
views

Say that a linear order $L$ is a thicket iff $L$ is infinite, and for all elements $a,b,c_1,...,c_n\in L$ with $a<_Lb$ and $[a,b]_L$ infinite the following are equivalent:
$\{a,b\}\subseteq \...

3
votes

1
answer

113
views

Here's a fairly easy fact from point-set topology that I'm having trouble finding a reference for. Say $X$ is a total order satisfying the least-upper bound property, and $S$ is a closed subset of it....

6
votes

0
answers

178
views

We study visualizations of attractors, which occur in chaotic dynamic systems, and for a few years trying to prove or refute Conjecture [3]. It has an equivalent formulation in terms of order theory, ...

6
votes

1
answer

213
views

Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...

3
votes

1
answer

140
views

Consider the set of finite sequences (of bounded length $\leq k$, if necessary) whose elements are taken from some finite alphabet $\Sigma$. We define a partial order on this set so that
$X = (X_1,...,...

9
votes

1
answer

225
views

The title says it all:
Let $E$ be a Banach lattice, which is isomorphic to a Hilbert space (as normed spaces). Is there an equivalent Hilbert norm on $E$, which still makes it a Banach lattice with ...

16
votes

1
answer

873
views

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 ...

1
vote

0
answers

95
views

How many permutations are there on the set $\{1,2, \cdots, n\}$ ($n\geq 3$), such that any three elements are not in increasing or decreasing order? For example, for $n=3$ we have $(1,3,2), (2,1,3), (...

1
vote

0
answers

79
views

In my context, I encounter a lot of partial orders with the distinguished property that the order is total on connected components. Equivalently, they satisfy the condition
$$x \le y,z \enspace \lor \...

11
votes

1
answer

363
views

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.
...

2
votes

1
answer

197
views

Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S_1$ and $S_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\...

9
votes

2
answers

445
views

It is well-known that the order type of any countable non-standard model of arithmetic $\mathfrak{A}$ is $\omega+(\omega^*+\omega)\eta$. My question is what could be said about the order types of ...

4
votes

1
answer

175
views

In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is ...

1
vote

1
answer

148
views

A nice theorem of Dushnik and Miller (from 1940) states that if $(\Omega,\leq)$ is a countable total order, then either
there is an element $\omega \in \Omega$ such that $(\Omega \setminus \{\omega\}...

8
votes

1
answer

282
views

By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following ...

3
votes

0
answers

41
views

Given a finite set $S$ with $n$ elements, and a fixed small $k$ (say $k=3$), how to find $k$ linear orders $\leq_1, \dots, \leq_k$ on $S$, such that the number of feasible subsets of $S$ is ...

3
votes

2
answers

133
views

This question is motivated by this problem of Dominic van der Zypen.
Problem. Let $X=\prod_{i=1}^nX_i$ be the Tychonoff product of linearly ordered compact Hausdorff spaces $X_1,\dots,X_n$. Is it ...

14
votes

1
answer

906
views

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 ...

4
votes

1
answer

135
views

In their paper "Theories with recursive models" [1] Lerman and Schmerl used a version of Kruskal's tree theorem about finite n-augmented trees.
An n-augmented tree is a tree T together with $n$ unary ...

2
votes

1
answer

280
views

I've found myself looking at a structure $\mathbb{M}$ whose important properties are:
$\mathbb{M}$ is a discretely ordered additive monoid.
$\mathbb{M}$ has a least element, and this least element is ...

9
votes

3
answers

565
views

Can there be a function $m(a,b)$ that is both associative and a mean, i.e.,
$\min (a,b) \leq m(a,b) \leq \max (a,b)$? The obvious solutions are $m(a,b) = \max(a,b)$ or $\min(a,b)$, but are there ...

6
votes

1
answer

410
views

Consider a complete first order theory $T$ whose language contains a binary predicate $\leq$. Assume that $T$ has an uncountable model that is well-ordered by $\leq$ so that this question isn't stupid ...

5
votes

1
answer

119
views

A difference between finite geometries and (e.g.) Euclidean space is that "lines" in finite geometries are unordered subsets of the universe, while "lines" in Euclidean space are ordered subsets of ...

4
votes

2
answers

209
views

This follows up on Incomparable dense linear orderings extending $\langle \mathbb{R},< \rangle$
and hopefully sparks more discussion.
Where $a<b$, say that the four “types” of nonempty bounded ...

7
votes

1
answer

257
views

Where $a<b$, say that the four “types” of non-empty bounded intervals are:
$(a,b)$, $[a,b]$, $(a,b]$, and $[a,b)$.
Let $\langle X,< \rangle$ and $\langle Y,< \rangle$ be dense linear ...

4
votes

1
answer

166
views

Notation: Let $L_1,L_2,...$ be linearly ordered sets.
$L_{i}^{-1}$ denotes the reverse linear order of $L_{i}$,
$L_1+L_2$ denotes the sum of linear orders, i.e. the disjoint union $L_1\cup L_2$ with ...

11
votes

0
answers

372
views

Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following?
$<$ is a linear order on a definable subset;
$\phi$ is $\aleph_1$-...

2
votes

1
answer

101
views

Is there a necessary and sufficient condition for a linearly ordered topological space to be pseudo-metrizable? (by a pseudo-metric, I mean a map $X\times X\rightarrow\mathbb{R}$ in which all the ...

3
votes

1
answer

162
views

If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$.
I am trying to prove the following:
If $(M,+,.,0,1)$ is a model of open induction, (or ...

18
votes

0
answers

765
views

I'd like to consider the principle asserting that the surreal
number line is universal for all class linear orders, or in other
words, that every linear order (including proper-class-sized)
linear ...

1
vote

0
answers

80
views

Suppose we have a partial order (efficiently computable), $\leq$, on $\mathbb{R}^n$, a set $S \subset \mathbb{R}^n$, and let $\rho$ be the standard Euclidean metric. We want to find a set $S^\prime = ...

4
votes

2
answers

465
views

Let $V$ be a real vector space. It is well known that, given a totally ordered basis of $V$ (say $(b_i)_{i\in I}$ where $I,<$ is totally ordered), $V$ is totally ordered by the lexicographic ...

19
votes

0
answers

748
views

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 $...

3
votes

4
answers

848
views

Q1.
Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...

1
vote

0
answers

205
views

A linear order $(L, <)$ is $\lambda$-transitive iff any order-preserving bijection between sets of size $\lambda$ can be extended to an order automorphism of $L$.
For $\lambda < \aleph_0$, $2$-...

3
votes

1
answer

279
views

A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group.
Is it true that every totally ordered ...

3
votes

1
answer

327
views

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm.
My problem is: I have M auctions and in each auction I have N ...

2
votes

3
answers

390
views

During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square.
I would really like to find examples of spaces like ...

4
votes

2
answers

485
views

Whenever $\kappa$ is an infinite cardinal number, write $L(\kappa)$ for the powerset of $\kappa$ ordered lexicographically. (Where the "$L$" stands for linear order.) Furthermore, write $B(\kappa)$ ...

2
votes

1
answer

180
views

A lot has been written about the arithmetic of ordinal numbers. However, we can also do arithmetic with linearly ordered sets.
Question. Is there an article or book where I can learn the basics of ...

7
votes

2
answers

773
views

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 ...

12
votes

0
answers

474
views

This is a question which has been bothering me now for quite some time. I've talked to a number of people about it, and we've shown that a few basic ideas can't work, but other than that haven't made ...

14
votes

3
answers

1k
views

I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and ...

4
votes

0
answers

204
views

My colleague Matthias Baaz is looking for a reference for the following question (or possibly theorem):
Let T be the "theory of pairs of finite linear orders". That is, consider all finite ...

15
votes

2
answers

502
views

A Suslin line is a linear order $L$ which is dense with no endpoints, complete, and ccc but not separable. I'm wondering what kind of order-preserving maps there are from $L$ into $L$. Specifically,
...

21
votes

2
answers

1k
views

(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:
$...

3
votes

1
answer

205
views

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order?
The answer is affirmative on abelian groups, where being torsion-free is ...