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Graph automorphism of a finite modular lattice. (English) Zbl 0953.06011

The author obtained the following result. Let \(L\) be a finite modular lattice. Then any automorphism of the unoriented graph corresponding to \(L\) is a lattice automorphism if and only if no direct factor of \(L\) having at least two elements is self-dual.
Thus the paper gives a partial solution of a problem concerning unoriented graphs of finite lattices (Problem 6 in [G. Birkhoff, Lattice theory, AMS, Providence, R.I. (1967; Zbl 0153.02501)]).
Reviewer: V.Slavík (Praha)

MSC:

06C05 Modular lattices, Desarguesian lattices

Citations:

Zbl 0153.02501
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References:

[1] G. Birkhoff: Lattice Theory. Second Edition, Providence, 1948. · Zbl 0033.10103
[2] G. Birkhoff: Lattice Theory. Third Edition, Providence, 1967. · Zbl 0153.02501
[3] J. Jakubík: On graph isomorphism of modular lattices. Czechoslovak Math. J. 4 (1954), 131-141.
[4] J. Jakubík, M. Csontóová: Convex isomorphisms of directed multilattices. Math. Bohemica 118 (1993), 359-379. · Zbl 0802.06008
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