# Tag Info

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### 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 ...
• 11.7k
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 ...
• 72.2k
Accepted

• 41.1k
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### 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 ...
• 9,047

### 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 ...
• 60.6k

### locally incomparable dense linear orderings extending $\langle \mathbb{R}, < \rangle$

Let me prove at least that it is consistent with ZFC that there are two linear orders as you requested. Work in ZFC plus the assumption that $2^{\omega_1}>\omega_3^L$, but $\omega_2=\omega_2^L$ ...

### 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 $\omega_1$ (...
• 1,355
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 $i$, ...
• 2,808
1 vote
Accepted

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 $((x,y,z),a,b)\in ... • 22.6k 1 vote Accepted ### Order-embeddability of${\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}$by induction ... • 4,501 1 vote ### What is the dimension of a subspace of the product of$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 ...
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### 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 ...
• 6,681

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