# Questions tagged [linear-orders]

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### Closed set in product topology implies convergence of monotonic sequences

Let $(X,\succsim)$ be a metrizable and connected, totally ordered topological space with the order topology. Let $\succeq$ be another order relation on $X \times X$ such that the two orders are ...
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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 $\lambda < \aleph_0$, $2$-...
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### Normal subgroup of a totally ordered group

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 ...
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### reduction to np hard ordering problem

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 ...
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### Compact, densely ordered spaces

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 ...
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### Is it consistent with ZFC that $\mathrm{dv}(\kappa) = \kappa$ for all infinite cardinal numbers $\kappa$?

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 $B(\kappa)$ ...
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### Linearly ordered set arithmetic: reference request

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 ...
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### 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 ...
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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 ...
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### Order homomorphism functions on $\omega_1$

This question has now been published in a math journal, see update at the bottom. I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains ...
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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 ...
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### Can a Suslin line be 2-entangled?

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, ...
(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: $... 1answer 180 views ### Extensions of partial orders to linear orders on (nonabelian) groups If$G$is a group with a (left) linear order, does every (left) partial order on$G$extend to a (left) linear order? The answer is affirmative on abelian groups, where being torsion-free is ... 2answers 522 views ### Extending a partial order while preserving an automorphism It is well known that if$(P, \leq)$is a partial order then$\leq$can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by ... 3answers 218 views ### Maximal chains in a quasi-order of linear order types Let$\mathcal{T}_\kappa$be the set of all linear order types of cardinality$\kappa$. Let$\prec$denote a binary relation on$\mathcal{T}_\kappa$representing embeddability of order types (note that ... 3answers 1k views ### Is it possible to construct an infinite subset of$\Bbb R$that is not order isomorphic to any proper subset of itself? Is it possible to construct an infinite subset of$\Bbb R$that is not order isomorphic to any proper subset of itself? 2answers 557 views ### How many subsets of$\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}$? I can see that results in both ... 2answers 309 views ### Is it possible to reconstruct an order type from its initial segments? Suppose$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$. Is it ... 1answer 270 views ### Rotation-invariant strict-inclusion-preserving preorderings on subsets of the circle Say that a preordering$\le$on a set of subsets of some space preserves strict inclusion provided that$A\lt B$whenever$A\subset B$(where$A\lt B$iff$A\le B$and$B \not\le A$). Let the space ... 1answer 485 views ### Characterizing$\omega_1$-like dense linear orderings I recently came upon the following theorem which was attributed to J. Conway: For each$A\subset \omega_1$, let$\Phi(A)$be a linear ordering of type$\sum_{\alpha<\omega_1} \tau_\alpha$, where$\...
I'm trying to find the correct term for a specific kind of totally ordered space: Let $S$ be a totally ordered space with strict total order $<$. Property: For any two $s_{1}$ and $s_{2}$ in $S$ ...