Timeline for Characterizing bases of 0-1-vectors in $\mathbb{R}^n$ in terms of their partial order

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Jun 23, 2014 at 10:50	comment	added	Max Flow		Thanks for answering this question. I suppose the problem is more interesting if we consider, in your 2nd example, that the two (isomorphic) posets are different subposets of the lattice $L$. Can we characterize affine independence of $V$ in $\mathbb{R}^n$ by a property of the subposet $(V,\preceq)$ of the lattice $L$? Consider, for instance, a simple necessary condition: If there exist $U \subseteq V$ and $v \in V \setminus U$ such that $\sup U = v$ and $\forall u,u' \in U: \inf(u,u') = \emptyset$ then $U$ is lin. dependent because $\sum_{u \in U} u = v$. Is there an if-and-only-if-condition?
May 15, 2014 at 18:47	comment	added	Emil Jeřábek		Well, $\{(0,0,0),(0,0,1),(0,1,0),(1,1,1)\}$ is affinely independent, $\{(0,0,0),(0,0,1),(0,1,0),(0,1,1)\}$ is dependent, and they have the same poset.
May 15, 2014 at 17:23	comment	added	Max Flow		Is the question as trivial if we consider affine independence instead of linear independence?
May 15, 2014 at 13:08	review	First posts
May 15, 2014 at 13:15
May 15, 2014 at 13:01	comment	added	Emil Jeřábek		No. Already for $n=1$, the posets for $V=\{0\}$ and $V=\{1\}$ are isomorphic.
May 15, 2014 at 12:57	history	edited	Max Flow	CC BY-SA 3.0	edited title
May 15, 2014 at 12:51	history	asked	Max Flow	CC BY-SA 3.0