### 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 | |

Publication status | Published - Jun 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Logic

### Cite this

*Studia Logica*,

*83*(1-3), 111-131. https://doi.org/10.1007/s11225-006-8300-x

**The Jónsson-Kiefer property.** / Adaricheva, K.; McKenzie, R.; Zenk, E. R.; Maróti, M.; Nation, J. B.

Research output: Contribution to journal › Article

*Studia Logica*, vol. 83, no. 1-3, pp. 111-131. https://doi.org/10.1007/s11225-006-8300-x

}

TY - JOUR

T1 - The Jónsson-Kiefer property

AU - Adaricheva, K.

AU - McKenzie, R.

AU - Zenk, E. R.

AU - Maróti, M.

AU - Nation, J. B.

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

UR - http://www.scopus.com/inward/record.url?scp=33746116498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746116498&partnerID=8YFLogxK

U2 - 10.1007/s11225-006-8300-x

DO - 10.1007/s11225-006-8300-x

M3 - Article

VL - 83

SP - 111

EP - 131

JO - Studia Logica

JF - Studia Logica

SN - 0039-3215

IS - 1-3

ER -