All Questions

A function $f:P\to Q$ from a poset $(P,\le_P)$ to a poset $(Q,\le_Q)$ is an order-embedding if, for all $p,p'\in P$, $p\le_P p'$ if and only if $f(p)\le_Q f(p')$. We partially order the Cartesian ...

Let $(X, \le)$ be a partially ordered set. We call a subset $S \subseteq X$... ... a chain if each two elements in $S$ are comparable with respect to $\le$ (in other words, $S$ is linearly ordered ...

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 = ...

I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and ...

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 ...