Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T13:03:40.846Z Has data issue: false hasContentIssue false
  • English
  • Français

𝔪-Stone Lattices

Published online by Cambridge University Press:  20 November 2018

B. A. Davey
B. A. Davey*
Affiliation:
Monash University, Clayton, Australia
Rights & Permissions [Opens in a new window]

Extract

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.

A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.

Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.

In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1027 - 1032
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Chen, C. C. and G., Grätzer, Stone lattices. I: construction theorems, Can. J. Math. 21 (1969), 884–894.Google Scholar
2. Frink, O., Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505–514.Google Scholar
3. Kist, J., Compact spaces of minimal prime ideals, Math. Z. III (1969), 151-158.Google Scholar
4. Mitchell, B., Theory of categories (Academic Press, New York, 1965).Google Scholar
5. Peercy, D. E., Complexes and the complete Baer extension of a commutative ring (to appear).Google Scholar
6. Speed, T. P., A note on commutative semigroups, J. Austral. Math. Soc. 8 (1968), 731–736.Google Scholar
7. Speed, T. P., Some remarks on a class of distributive lattices, J. Austral. Math. Soc. 9 (1969), 289–296.Google Scholar
8. Speed, T. P., A note on commutative Baer rings III (to appear).Google Scholar
9. Swan, G., The theory of sheaves (University of Chicago Press, Chicago, 1964).Google Scholar