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The Jónsson-Kiefer Property

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Abstract

The least element 0 of a finite meet semi-distributive lattice is a meet of meet-prime elements. We investigate conditions under which the least element of an algebraic, meet semi-distributive lattice is a (complete) meet of meet-prime elements. For example, this is true if the lattice has only countably many compact elements, or if |L| < 2ℵ0, or if L is in the variety generated by a finite meet semi-distributive lattice. We give an example of an algebraic, meet semi-distributive lattice that has no meet-prime element or join-prime element. This lattice L has |L| = |LC| = 2ℵ0 where Lc is the set of compact elements of L.

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Correspondence to K. Adaricheva.

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Dedicated to the memory of Willem Johannes Blok

AMS subject classification: 06B05

While working on this paper, the first author was supported by the INTAS grant no. 03-51-4110, the second author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877, and the third author was supported by the US National Science Foundation grant no. DMS0245622.

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Adaricheva, K., Mckenzie, R., Zenk, E.R. et al. The Jónsson-Kiefer Property. Stud Logica 83, 111–131 (2006). https://doi.org/10.1007/s11225-006-8300-x

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  • DOI: https://doi.org/10.1007/s11225-006-8300-x

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