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

Year, pages: 2018, 1321 - 1326

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 \textbf{37} (1975), 79--82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving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.

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

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