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Pseudo-complemented modular semilattices

Published online by Cambridge University Press:  09 April 2009

William H. Cornish
William H. Cornish
Affiliation:
The Flinders University of South Australia, Bedford Park, South Australia 5042
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Chen and Grätzer [3], [4] have had great success in describing the properties of Stone lattices by way of representing them as triples. Their triple representation has recently been generalized to distributive pseudo-complemented lattices by Katriňák [6]. By varying the approach slightly Katriňák [5] has been able to obtain a triple representation for distributive pseudo-complemented semilattices that enabled him to characterize semilattices from distributive pseudo-complemented semilattices through to Stone lattices and Brouwer lattices in an elegant unified manner.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 2 , September 1974 , pp. 239 - 251
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Balbes, R., ‘A representation theory for prime and implicative lattices’, Trans. Amer. Math. Soc. 136 (1969), 261–267.CrossRefGoogle Scholar
[2]Birkhoff, G., Lattice theory. (Amer. Math. Soc. Coil. Publ. Vol. 25, 3rd Ed. 1967).Google Scholar
[3]Chen, C. C. and Grätzer, G., ‘Stone lattices. I, Construction Theorems’, Canad. J. Math. 21 (1969), 884–894.CrossRefGoogle Scholar
[4]Chen, C. C. and Gratzer, G., ‘Stone lattices. II, Structure Theorems’, Canad. J. Math. 21 (1969), 895–903.CrossRefGoogle Scholar
[5]Katriňák, T., ‘Die Kennzeichnung der distributiven pseudo-komplementären Halbverbände’, J. fur reine undangew. Math. 241 (1970), 160–179.Google Scholar
[6]Katriňák, T., ‘Über eine Konstruktion der distributiven pseudo-komplementären Verbände’, (prepublication copy).Google Scholar
[7]Ore, O., ‘Theory of equivalence relations’, Duke Math. J. 9 (1942), 573–627.CrossRefGoogle Scholar
[8]Rhodes, J. B., ‘Modular semilattices’, Abstract 672–659, Notices Amer. Math. Soc. 17 (1970), 272.Google Scholar
[9]Rhodes, J. B., ‘A characterization of modular semilattices by their retracts’, Abstract 678–A4, Notices Amer. Math. Soc. 17 (1970), 930.Google Scholar
[10]Rhodes, J. B., ‘Modular and distributive semilattices’, (prepublication copy).Google Scholar
[11]Schmidt, E. T., ‘Zur Charakterisierung der Kongruenzverbände der Verbände’, Mat. Cäsop. 18 (1968), 3–20.Google Scholar
[12]Varlet, J. C., ‘Modularity and distributivity in partially ordered groupoids’, Bull. Soc. Roy, Sci. Liége, 38 (1969), 639–648.Google Scholar