# Questions tagged [order-theory]

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### Embedding of Coxeter groups into product of trees - Relationship between partial orders

Let $(W,S)$ be a Coxeter system. A result by Ranishnikov and Januszkiewicz (see Every Coxeter group acts amenably on a compact space for the construction) states that $W$ can be isometrically embedded ...
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### Reference request: a survey of (linear) Krein-Rutman theory

I'm looking for a survey article or book chapter where a rather exhaustive treatment of the Krein-Rutman theory of positive linear operators an ordered Banach spaces is given. Motivation. Some ...
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### Explicit lifting characterization of complete lattices among posets?

It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property ...
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### Dubious matrix monotonicity

Coming from a problem in game theory, I arose at some dubious monotonicity like property for matrices of the following art. Let $H=\lbrace h\in\mathbb{R}^{n}\colon h_{1}+\dots+h_{n}=0\rbrace$. I'm ...
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### Is this ordering on the set of all covers of $\omega$ a (complete) lattice?

Let ${\frak C} \subseteq {\cal P}({\cal P}(\omega))$ be the collection of all covers of $\omega$ (that is, ${\cal C} \in {\frak C}$ iff $\bigcup {\cal C} = \omega$.) We define the following binary ...
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### How can you order a free group?

A left order on a (discrete) group $G$ is a total order on $G$ satisfying $\forall g,h,k \in G: g < h \implies kg < kh$. A right order is defined symmetrically, and a biorder is an order that is ...
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### Explicit calculation of the width of a product of chains (i.e. maximal rank size)

Given a poset $P$, I am interested in the width (size of the maximal antichain) of $\mathcal{O}(P)$, i.e. the poset of downsets in $P$, ordered by inclusion. As this is rather difficult, I'm starting ...
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### When does a graph underlie the Hasse diagram of a poset?

For any finite poset $P=(X,\leq)$ there is a graph $G$ underlying its Hasse diagram $H=(X,\lessdot)$, so that $V(G)=X$ and $E(G)=\{\{u,v\}:u\lessdot v\}$. With that said, is it possible to ...
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### Generalization of the linear extension theorem to directed acyclic graphs

Using Zorn's lemma one can prove a generalization of the order extension theorem, that states any acyclic digraph is always contained in another acyclic unilaterally connected digraph on the same ...
### Minimizing the set of “faulty” edges in a map between the vertex sets of $2$ graphs
The starting point of this question is the fact that for some simple, undirected graphs $G, H$ there is no graph homomorphism $f:G\to H$. This is the case for instance if $\chi(G)>\chi(H)$. ...