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Keywords:
modular lattices; prime quotients; order-dense quotients; valuation; discrete valuation
modular lattices; prime quotients; order-dense quotients; valuation; discrete valuation
Summary:
It is well-known that there exist infinite modular lattices possessing no non-trivial valuations. In this paper a class $\Cal K$ of modular lattices is defined and it is proved that each lattice belonging to $\Cal K$ has a nontrivial valuation. Next, a result of $G$. Birkhoff concerning valuations on modular lattices of finite length is generalized.
References:
[1] G. Birkhoff: Lattice Theory. Providence 1967. MR 0227053 | Zbl 0153.02501
[2] G. Grätzer: General Lattice Theory. Akademie Verlag, Berlin, 1978. MR 0504338
[3] B. Monjardet: Metrics on partially ordered sets - a survey. Discrete Math. 35 (1981), 173-184. DOI 10.1016/0012-365X(81)90206-5 | MR 0620670 | Zbl 0463.46016
[4] E. T. Schmidt: Über die Kongruenzverbände der Verbände. Publ. Math. Debrecen 9 (1962), 245-256. MR 0151405 | Zbl 0178.33902
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