Questions tagged [linear-orders]
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22
questions with no upvoted or accepted answers
19
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765
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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 $...
18
votes
0
answers
854
views
Is the universality of the surreal number line a weak global choice principle?
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 ...
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.
...
12
votes
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492
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Do all linear orders in this class have computable copies?
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 ...
11
votes
0
answers
423
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Example of $\aleph_1$-categorical linear order
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 $\aleph_1$-...
10
votes
0
answers
361
views
Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?
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 ...
6
votes
0
answers
188
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A representation of a partial order by a slowly changing sequence of linear orders
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, ...
5
votes
0
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125
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Which monomials are "leadable"?
Question: Let $k$ be a field, let $f \in k[t_1,\ldots,t_N]$ be a nonzero polynomial. Which monomials
$m_a = t_1^{a_1} \cdots t_N^{a_n}$ appearing in $f$ are leadable in the sense that they are the ...
5
votes
0
answers
189
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Given a totally ordered system of Banach spaces, can we we always change the norms to get isometric embeddings?
Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...
4
votes
0
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205
views
Is there a ${\bf 0'}$-computable linear order with "all intervals wild"?
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 \...
4
votes
0
answers
211
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The theory of two finite linear orders
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 ...
3
votes
0
answers
69
views
What are all the order types of maximal chains of $\Delta^0_2$ sets?
A set of natural numbers is $\Delta^0_2$ if it’s computable from the halting set. Consider the quasi-order/pre-order of all $\Delta_0^2$ sets ordered by $m$-reduction, or equivalently consider the ...
3
votes
0
answers
44
views
Maximal number of sets, obtained as intersections of semiintervals of $k$ linear orders
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 ...
2
votes
0
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79
views
Ordered vector space that can be embedded into its bidual
We say that an ordered vector space $(V, \ge)$ (over $\mathbb{R}$) is "bidual embeddable" (I made up this name, not sure whether this concept already exists) if for every $x \in V$, if $x$ ...
2
votes
0
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60
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Countable highly order-transitive subgroups of $\mathrm{Aut}(\mathbb{Q},\leq)$
Consider $A := \mathrm{Aut}(\mathbb{Q},\leq)$, the group of order-automorphisms of $(\mathbb{Q},\leq)$. Call a subgroup $U$ highly order-transitive if for any two finite ordered sequences $s_1$ and $...
2
votes
0
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97
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Closed images of linearly ordered spaces
Is there a description of the class of continuous closed images of linearly ordered spaces?
1
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0
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180
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A variant of Buchholz's ordinal notation
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 $(\mathcal{OT}, \prec)$ ...
1
vote
0
answers
99
views
About a type of permutations
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
85
views
Name for partial orders which are total on connected components
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 \...
1
vote
0
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83
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reconstructing a linear order corrupted by noise
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 = ...
1
vote
0
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209
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Existence of $\lambda$-transitive linear orders for $\lambda \geq \aleph_0$
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 $\lambda < \aleph_0$, $2$-...
0
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0
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24
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Reference for tree of bad sequences of WPO
I'm looking for a reference to give in Wikipedia for the following result: Let $X$ be a WPO. Let $T_X$ be the tree of bad sequnces of $X$, and let $o(X)$ be the ordinal height of the root of $T_X$. ...