• DOI:10.1006/AAMA.1997.0536
• Corpus ID: 121414616

Three Notes on Distributive Lattices

@article{Wild1998ThreeNO,
  title={Three Notes on Distributive Lattices},
  author={Marcel Wild},
  journal={Advances in Applied Mathematics},
  year={1998},
  volume={20},
  pages={44-49}
}

• M. Wild
• Published 1998
• Mathematics
• Advances in Applied Mathematics

The present paper comprises three rather independent notes. The first and second notes are about finite distributive lattices. We hasten to add that the title is a bit unprecise; the third note is actually about finite meet semidistributive lattices. In Section 1 an example of a self-dual distributive lattice, which does not allow for a polarity is given. This answers a question of Kamara. Another question of Metropolis, Rota, and Stein is settled as well. Both problems fit well into the framework of De Morgan algebras. Section 2 gives an elementary proof in the distributive case of a theorem of Reuter on proper matchings in modular lattices. As to Section 3, by a theorem of Reiterman, each pseudovariety of finite algebras can be described by implicit operations. We exhibit a ‘‘natural’’ implicit operation for the pseudovariety of finite SD -lattices. n