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Introduction

I am studying a problem in which the antichains of a poset are of key importance. They are naturally geometrically embedded as vectors in the space $\mathbb{R}^P$, where $P$ is the poset, and I want to consider all the subsets of these vectors that are linearly independent in $\mathbb{R}^P$. As such I already have a matroid representation of the antichains but I don't know what the corresponding independent sets are. My feeling is that there should be a natural description of independence in terms of the poset itself but I don't know what it is. I should mention that this question is very far from my usual area of research, so there may be a simple answer which I am not aware of.

Setup and Embedding

Let $(P, <)$ be a finite poset. One may make any assumptions about it that are necessary, I usually am thinking about a finite square box in $\mathbb{Z}^2$ with the componentwise order. Letting $A$ be an antichain of $P$, I then embed $A$ into $\mathbb{R}^P$ via its ``indicator function'' $$ \mathbf{1}_A(\mathbf{v}) = \begin{cases} 1, \quad \mathbf{v} \in A, \\ 0, \quad \mathbf{v} \not \in A. \end{cases} $$ On the vector space side a collection of these vectors is said, as usual, to be an independent set if the vectors are linearly independent. Now I'm looking at the "inverse problem": given this representation, what is the corresponding notion of an independent set for the antichains? Ideally I would like a description in terms of the poset itself.

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