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A geometric approach to representation of a partially ordered set. (English. Russian original) Zbl 0656.06003

Vestn. Leningr. Univ., Math. 21, No. 1, 11-15 (1988); translation from Vestn. Leningr. Univ., Ser. I 1988, No. 1, 12-14 (1988).
Let P be a finite ordered set, k a field. A representation of P over k is an isotonic mapping \(\pi\) of P into the lattice of subspaces of a vector space \(k^ n\). A representation \(\pi\) is called without centre if the lattice generated by \(\{\) \(\pi\) (x): \(x\in P\}\) has trivial centre. The vector (dim \(\pi\) (x): \(x\in P)\) is called a dimension vector of \(\pi\). Two representations are combinatorially equivalent if they have the same dimension vector; they are projectively equivalent if there exists a projective mapping twinning the corresponding subspaces. For a given finite set \(\alpha\) of subspaces of \(k^ n\) let u(\(\alpha)\) be the set of all linear hulls of subsets of \(\alpha\) and \(\ell (\alpha)\) be the set of all intersections of subsets of \(\alpha\). Further, put \(c_ 0(\alpha)=\alpha\), \(c_{2k+1}(\alpha)=u(c_{2k}(\alpha))\), \(c_{2k}(\alpha)=\ell (c_{2k-1}(\alpha))\). Two representations \(\pi_ 1\) and \(\pi_ 2\) of P are combinatorially equivalent of degree k if \(c_ k(\pi_ 1(P))\) and \(c_ k(\pi_ 2(P))\) have the same dimension vector. The main result of the paper is the following one: Two representations without centre of a finite ordered set P are projectively equivalent iff they are combinatorially equivalent of degree k for all k.
Reviewer: V.Novák

MSC:

06A06 Partial orders, general
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