### 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 |
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Pages (from-to) | 111-131 |

Number of pages | 21 |

Journal | Studia Logica |

Volume | 83 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Jun 1 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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## Cite this

*Studia Logica*,

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