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Keywords:
lower continuous lattices; strongly dually atomic lattices; semimodular and atomic lattices
lower continuous lattices; strongly dually atomic lattices; semimodular and atomic lattices
Summary:
For lattices of finite length there are many characterizations of semimodularity (see, for instance, Grätzer [3] and Stern [6]–[8]). The present paper deals with some conditions characterizing semimodularity in lower continuous strongly dually atomic lattices. We give here a generalization of results of paper [7].
References:
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[2] Crawley, P., Dilworth, R. P.: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs (N.J.), 1973.
[3] Grätzer, G.: General Lattice Theory. Birhäuser Basel, 1978. MR 0509213
[4] Richter, G.: The Kuroš-Ore Theorem, finite and infinite decompositions. Studia Sci. Math. Hungar., 17(1982), 243-250. MR 0761540
[5] Stern, M.: Exchange properties in lattices of finite length. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 31 (1982), 15-26. MR 0693283 | Zbl 0548.06003
[6] Stern, M.: Semimodularity in lattices of finite length. Discrete Math. 41 (1982), 287-293. MR 0676890 | Zbl 0655.06006
[7] Stern, M.: Characterizations of semimodularity. Studia Sci. Math. Hungar. 25 (1990), 93-96. MR 1102200 | Zbl 0629.06007
[8] Stern, M.: Semimodular Lattices. B. G. Teubner Verlagsgesellschaft, Stuttgart-Leipzig, 1991. MR 1164868 | Zbl 0957.06008
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