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Licensed Unlicensed Requires Authentication Published by De Gruyter November 20, 2018

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

Marcin Łazarz EMAIL logo
From the journal Mathematica Slovaca

Abstract

J. Jakubík noted in [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving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 S7. 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.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] cannot be strengthened is analogous way.

  1. (Communicated by Miroslav Ploščica )

Acknowledgement

I am grateful to the anonymous referee for his important remarks which enabled me to significantly improve the first version of the paper.

References

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Received: 2016-10-31
Accepted: 2018-03-05
Published Online: 2018-11-20
Published in Print: 2018-12-19

© 2018 Mathematical Institute Slovak Academy of Sciences

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