31
votes
Accepted
Characterizing $\mathbf{R}$ as an ordered group
The linearly ordered group $(\mathbb{Z},+,\le)$ is a counterexample, but that is probably not what the OP had in mind. To give a detailed description of the situation, let us use the following ...
- 12.5k
22
votes
Accepted
Suprema of directed sets
Yes, a poset that has suprema of all chains also has suprema of all directed sets. This is known, and I vaguely recall seeing it attributed to Solovay. The proof consists of showing, by induction on ...
- 73.1k
12
votes
Accepted
Associative mean
Yes there are more such functions, even if we require them to be symmetric, monotonic and continuous. For example pick any $C$ and take
$m_C(a,b)=\min(a,b)$ if $a,b\ge C$,
$m_C(a,b)=\max(a,b)$ if $a,...
- 5,103
12
votes
Associative mean
Let $(L, <)$ be any linear order. Fix an arbitrary well-order $\sqsubset$ on $L$. For any $a\le b$ define $m(a,b)=m(b,a):= $ the $\sqsubset$-least element in the interval $[a,b]$.
Then $m$ is ...
- 14k
9
votes
Accepted
Upward Löwenheim–Skolem theorem for well-ordered models with/without measurable cardinals
A very nice collection of questions.
Here are a few things one can say to get started.
If $\kappa$ is a measurable cardinal and $T$ has a well-ordered
model of size at least $\kappa$, then it has ...
- 236k
6
votes
Classifying the endofunctors of the category $\Delta$ of finite linear orders
See Edgewise subdivision and simple maps by Knut Berg (supervised by me), Generalized edgewise subdivisions by Katerina Velcheva (supervised by Clark Barwick) and the earlier MathOverflow question ...
- 9,263
6
votes
Accepted
Is there an explicit linear extension for the subsequence partial order?
Note that two words of the same lengths are comparable if and only if they are equal. So you can order the words in the following way:
$$X\prec^* Y\iff |X|<|Y|\text{ or } (X<_{\rm Lex}Y \text{ ...
- 39.7k
6
votes
Accepted
Ordinal notations within non-standard models of arithmetic
I'll provide a description of $(\alpha)^{\mathfrak{A}}$ in the case of countable $\mathfrak{A}$. I base my answer on observations by Emil Jeřábek (see discussion below the initial question), but of ...
- 5,821
6
votes
Characterization of Archimedean linearly ordered monoids
By a classic result of Alimov (1950), a fully ordered semigroup $S$ is order isomorphic to a subsemigroup of the additive group of real numbers if and only if it is cancellative and contains no ...
- 6,453
5
votes
Accepted
Is there literature on finite geometries with ordered lines?
Yes, this has been studied and is indeed known as ordered geometry or the study of betweenness spaces:
https://en.m.wikipedia.org/wiki/Ordered_geometry
- 24.8k
5
votes
Accepted
Fundamental theorem of linear orders
As already observed, you may as well assume $(\Omega,\leq)$ is $(\mathbb{Q},\leq)$. In this case, of course one condition on $\Omega_1$ and $\Omega_2$ ensuring your desired property is finiteness.
...
- 3,274
4
votes
Ordinal notations within non-standard models of arithmetic
I spent some time thinking about order-types of models of arithmetic, and of order-types of structures they interpret (my PhD). I think there are several formulations of your question: (1) about ...
- 443
4
votes
Reference for tree of bad sequences of WPO
OK I found an answer myself. Blass and Gurevich (2008) fill in the missing details (and reprove the result in its entirety).
Andreas Blass and Yuri Gurevich, Program termination and well partial ...
- 1,174
4
votes
Is a Banach lattice isomorphic to a Hilbert space in fact a Hilbert lattice?
Yes. This is a consequence of the fact that semi normalized unconditionally basic sequences in a Hilbert space are Riesz bases. For a theorem quoting proof of what you want, note that that the lattice ...
- 31.5k
3
votes
Accepted
A closed subset of a Dedekind-complete order has subspace topology equal to order topology
While I agree that it's pretty direct to show, I was unable to find a reference for a proof of this fact myself (I thought it was in Willard, but I thumbed through my copy and failed to find it). So ...
- 1,322
3
votes
Ordering preference for two zero mean Gaussian outcomes
It seems like it is high time to handle this one. The main difficulty here is that there seems to be no conceptual reason for the inequality to be true: it just comes up valid before one numerical ...
- 61.9k
3
votes
Topology on set of "real lower bounds"
A slight modification of this space, where you let $t$ range over $[0,1]$ instead of $\mathbb R$, is known as the double arrow space or the split interval. You can learn more about it here or here.
- 18.5k
3
votes
Accepted
Permutations which respect a partial order
In more common language, the question is whether for each finite poset, any two linear extensions of the poset are related by a series of adjacent transposition that exchange incomparable elements.
...
- 24.2k
3
votes
Self-embeddings of uncountable total orders
I may be missing something, but it seems to me that the exact same statement holds for all infinite total orders.
Indeed, if $\mathbb Q$ does not embed in $\Omega$, then $\Omega$ is scattered, hence ...
- 47.1k
3
votes
Accepted
A strictly decreasing function between uncountable subsets of the reals
In Todorcevic's book "Partition Problems in Topology" I have found Proposition 8.4(c) saying that under OCA for any uncountable sets $X,Y$ of reals there exists a strictly increasing function $f:Z\to ...
- 41.8k
3
votes
Accepted
Name for this algebraic structure?
As Emil Jeřábek correctly suggests in the comments above, $\mathbb{M}$ is exactly the non-negative part of a discretely ordered group -- I humbly offer a proof to close the thread. The proof that the ...
- 10.1k
3
votes
Associative mean
Let us, indeed, describe "monotone, continuous, associative means" on $[0,1]^2$. I'll just write $a*b$ instead of $m(a,b)$ to make long expressions easier to comprehend. Let $c=0*1,d=1*0$. Then, for ...
- 61.9k
2
votes
Order homomorphism functions on $\omega_1$
The answer to the above question is no (there is no such mapping $\psi$).
The question asked above is an order-theoretic restatement of a question from general topology:
Question. Does $\omega_1$ (...
- 1,375
2
votes
Accepted
Kruskal's tree theorem and $\Pi_1$ sentences of linear orderings with finitely many constants
Towards a contradiction, suppose not. Then we can find a sequence $\phi_0, \phi_1, \dots$ from $\Phi$ such that for all $i$, $LO, \phi_0, \dots, \phi_i \not \vdash \phi_{i+1}$. So for each $i$, ...
- 3,073
1
vote
Accepted
Find an order-embedding of $S_3\times{\bf2}\times{\bf2}$ into ${\mathbb Z}^4$
We present an embedding into $\mathbb Z^3$. It is symmetric with respect to the premutation of the coordinates in $S_3$, so it suffices to show the images of the following elements $((x,y,z),a,b)\in ...
- 23.7k
1
vote
Accepted
Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$
Assume that $i : \alpha \to \mathbb{R}$ is an order-embedding of some ordinal $\alpha$ into $(\mathbb{R},<)$. We can modify $i$ to yield an order embedding $j : \alpha \to \mathbb{Q}$ by induction ...
- 4,667
1
vote
What is the dimension of a subspace of the product of $n$ linearly ordered compacta
In an old PhD-thesis "Finite products of locally compact ordered spaces" by J. van Dalen (Vrije Universiteit, Amsterdam) from 1972, I found (I could not find a paper with the result, so far, as I have ...
- 5,407
1
vote
Order homomorphism functions on $\omega_1$
I have been thinking about this question since it was bumped up. I am burned out now, so I thought I would post a few basic things that I noted.
First, as you noted that any function $f \in \mathcal ...
- 881
1
vote
An algebraically generated set of linear orders
Thanks to Goldstern for the reference. Very helpful! I finally found time to look through a copy of "Linear Orderings" by Joseph Rosenstein, which is a very nice book. It turns out that Rosenstein ...
- 7,193
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