Covering graphs of lattices. (English) Zbl 0606.06005

Lee, Jeh Gwon

Bull. Korean Math. Soc. 23, 39-46 (1986).

The following theorem is proved: Let L and L’ be lattices with isomorphic covering graphs. If L is geometric, then \(L\cong A\times B\) and \(L'\cong A^ d\times B\) for some sublattices A and B, where \(A^ d\) is the dual of A. Next, an analogous situation concerning modular lattices is dealt with (the case of modular lattices was investigated earlier by the reviewer [Czech. Math. J. 4(79), 131-141 (1954; Zbl 0059.026); ibid. 25(100), 240-246 (1975; Zbl 0314.06006)]).

Reviewer: J.Jakubík

MSC:

06B05	Structure theory of lattices
06C05	Modular lattices, Desarguesian lattices
06C10	Semimodular lattices, geometric lattices
05C99	Graph theory

Keywords:

geometric lattice; lattices with isomorphic covering graphs; modular lattices

Citations:

Zbl 0059.026; Zbl 0314.06006