# An extension of F. Šik's theorem on modular lattices

In: Mathematica Slovaca, vol. 68, no. 6
Marcin Łazarz
Detaily:
Rok, strany: 2018, 1321 - 1326
Kľúčové slová:
modular lattice, upper semimodular lattice, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice
O článku:
J. Jakubík noted in [Jakubík, J.: \textit{Modular Lattice of Locally Finite Length}, Acta Sci. Math. \textbf{37} (1975), 79--82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic to $S_7$. In the present paper we extend the scope of Šik's theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [Jakubík, J.: \textit{Modular Lattice of Locally Finite Length}, Acta Sci. Math. \textbf{37} (1975), 79--82] cannot be strengthened is analogous way.
Ako citovať:
ISO 690:
Łazarz, M. 2018. An extension of F. Šik's theorem on modular lattices. In Mathematica Slovaca, vol. 68, no.6, pp. 1321-1326. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0182

APA:
Łazarz, M. (2018). An extension of F. Šik's theorem on modular lattices. Mathematica Slovaca, 68(6), 1321-1326. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0182
O vydaní:
Publikované: 3. 12. 2018