Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-27T18:16:45.057Z Has data issue: false hasContentIssue false
  • English
  • Français

Stone Lattices. I: Construction Theorems

Published online by Cambridge University Press:  20 November 2018

C. C. Chen  and
G. Grätzer
C. C. Chen
Affiliation:
Queen's University, Kingston, Ontario
G. Grätzer
Affiliation:
Nanyang University, Singapore, Malaysia University of Manitoba, Winnipeg, Manitoba
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.

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 884 - 894
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Birkhoff, G., Lattice theory, third éd., Amer. Math. Soc. Colloq. Publ., Vol. 25 (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
2. Bruns, G., Ideal representations of Stone lattices, Duke Math. J. 32 (1965), 555–556.Google Scholar
3. Frink, O., Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505–514.Google Scholar
4. Grâtzer, G., A generalization of Stone's representation theorem for Boolean algebras, Duke Math. J. 30 (1963), 469–474.Google Scholar
5. Grâtzer, G. and Schmidt, E. T., On a problem of M. H. Stone, Acta Math. Acad. Sci. Hungar. 8 (1957), 455–460.Google Scholar
6. Nachbin, L., Une propriété caractéristique des algèbres booléiennes, Portugal. Math. 6 (1947), 115–118.Google Scholar
7. Nemitz, W. C., Implicative semi-lattices, Trans. Amer. Math. Soc. 117 (1965), 128–142.Google Scholar
8. Stone, M. H., Topological representations of distributive lattices and Brouwerian logics, Casopis Pest. Mat. Fys. 67 (1937), 1–25.Google Scholar
9. Varlet, J., On the characterization of Stone lattices, Acta Sci. Math. (Szeged) 27 (1966), 81–84.Google Scholar