Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T11:30:17.884Z Has data issue: false hasContentIssue false
  • English
  • Français

Atoms, Primes and Implicative Lattices

Published online by Cambridge University Press:  20 November 2018

C. S. Hoo
C. S. Hoo*
Affiliation:
University of Alberta, Edmonton, Alberta, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let L be an a-implicative semilattice. We obtain a characterization of those elements which cover a. This gives a characterization of atoms in pseudocomplemented semilattices, and leads to various results on primes and irreducibles in semilattices. As an application, we prove that in a complete, atomistic lattice L, the following are equivalent (i) L is implicative (ii) L is (2, ∞) meet distributive (iii) each element of L is a meet of primes.

Keywords

06D2003G0506D1503G10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 3 , 01 September 1984 , pp. 279 - 285
Copyright
Copyright © Canadian Mathematical Society 1984

References

[1] Balbes, R., A representation theory for prime and implicative semilattices, Trans. Amer. Math. Soc. 136 (1969), 261–267.Google Scholar
[2] Balbes, R. and Dwinger, P., Distributive lattices, University of Missouri Press, 1974.Google Scholar
[3] Birkhoff, G., Lattice theory, AMS. Coll. Pub. 25 (1961).Google Scholar
[4] Curry, H. B., Foundations of Mathematical logic, McGraw-Hill, New York, 1976.Google Scholar
[5] Grätzer, G., General Lattice Theory, Academic Press, New York, 1978.Google Scholar
[6] Hoo, C. S., Pseudocomplemented and implicative semilattices, Can. J. Math. 35 (1982), 423–437.Google Scholar
[7] Nemitz, W. C., Implicative semilattices, Trans. Amer. Math. Soc. 117 (1965), 128–142.Google Scholar
[8] Smith, D. P., Meet-irreducible elements in implicative lattices, Proc. Amer. Math. Soc. 34 (1972), 57–62.Google Scholar
[9] Varlet, J. C., Relative annihilators in semilattices, Bull. Austral. Math. Soc. 9 (1973), 169–185.Google Scholar