The Jónsson-Kiefer property

K. Adaricheva; R. McKenzie; E. R. Zenk; M. Maróti; J. B. Nation

doi:10.1007/s11225-006-8300-x

The Jónsson-Kiefer property

K. Adaricheva, R. McKenzie, E. R. Zenk, M. Maróti, J. B. Nation

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

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| < 2^{א. 0}, 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| = 2^א0 where L_c is the set of compact elements of L.

Original language	English
Pages (from-to)	111-131
Number of pages	21
Journal	Studia Logica
Volume	83
Issue number	1-3
DOIs	https://doi.org/10.1007/s11225-006-8300-x
Publication status	Published - Jun 2006

Keywords

Join semi-distributive lattice
Join-prime element
Meet semi-distributive lattice
Meet-prime element
Pseudo-complemented lattice

ASJC Scopus subject areas

Logic
History and Philosophy of Science

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title = "The J{\'o}nsson-Kiefer property",

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| < 2א. 0, 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.",

keywords = "Join semi-distributive lattice, Join-prime element, Meet semi-distributive lattice, Meet-prime element, Pseudo-complemented lattice",

author = "K. Adaricheva and R. McKenzie and Zenk, {E. R.} and M. Mar{\'o}ti and Nation, {J. B.}",

note = "Funding Information: ∗ 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. DMS–0245622. Copyright: Copyright 2006 Elsevier B.V., All rights reserved.",

year = "2006",

month = jun,

doi = "10.1007/s11225-006-8300-x",

language = "English",

volume = "83",

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AU - Adaricheva, K.

AU - McKenzie, R.

AU - Zenk, E. R.

AU - Maróti, M.

AU - Nation, J. B.

N1 - Funding Information: ∗ 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. DMS–0245622. Copyright: Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/6

Y1 - 2006/6

N2 - 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| < 2א. 0, 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.

AB - 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| < 2א. 0, 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.

KW - Join semi-distributive lattice

KW - Join-prime element

KW - Meet semi-distributive lattice

KW - Meet-prime element

KW - Pseudo-complemented lattice

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The Jónsson-Kiefer property

Abstract

Keywords

ASJC Scopus subject areas

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