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Some finiteness conditions in lattices—using nonstandard proof methods

Published online by Cambridge University Press:  09 April 2009

Matt Insall
Affiliation:
Department of Mathematics and StatisticsUniversity of Missouri at RollaRolla, Missouri 65401-0249, USA
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Abstract

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We discuss the application of nonstandard methods to local versions of certain lattice notions. In a particular case, we find that imposition of certain local conditions imply a surprising global one, namely boundedness of the given lattice.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Bandelt, H. H., ‘Tolerance relations on lattices’, Bull. Austral. Math. Soc. 23 (1981), 367381.CrossRefGoogle Scholar
[2]Birkhof, G., Lattice theory, Amer. Math. Soc. Colloquium Publications 25 (Providence, R.I., 1967).Google Scholar
[3]Chajda, I. and Zelinka, B., ‘Tolerances and convexity’, Czech. Math. J. 29 (104) (1979), 367381.CrossRefGoogle Scholar
[4]Chajda, I. and Zelinka, B., ‘A characterization of tolerance-distributive tree semilattices’, Czech. Math. J. 37 (112), (1987) No. 2.CrossRefGoogle Scholar
[5]Gehrke, M., Insall, M. and Kaiser, K., ‘Some nonstandard methods applied to distributive lattices’, Zeitschr. f. Math. Logik und Grundlagen d. Math., 1990, 123131.CrossRefGoogle Scholar
[6]Gratzer, G., Universal algebra (Springer-Verlag, New York, 1979).CrossRefGoogle Scholar
[7]Gratzer, G., General lattice Theory, Academic Press, Orlando, Fla., 1978.CrossRefGoogle Scholar
[8]Gonshor, H., ‘Enlargements contain various kinds of completions’, in: Proceedings of the 1972 Victoria Symposium on Nonstandard Analysis, Springer Lecture Notes in Mathematics 369 (1974), 6070.CrossRefGoogle Scholar
[9]Hurd, A. E. and Loeb, P. A., An introduction to nonstandard real analysis (Academic Press, Orlando, Fla., 1985).Google Scholar
[10]Hobby, D. and McKenzie, R., The Structure of Finite Algebras, Amer. Math. Soc. Series in Contemporary Mathematics, v. 76. (Providence, R.I., 1988).CrossRefGoogle Scholar
[11]Insall, M., ‘Nonstandard Methods and Finiteness Conditions in Algebra’, Zeitschr. f. Math. Logik und Grundlagen d. Math., 1991.CrossRefGoogle Scholar
[12]Luxemburg, W. A. J., ed., Applications of model theory to algebra, analysis and probability (Holt, Rinehart and Winston, New York, 1969).Google Scholar
[13]Robinson, A., Nonstandard analysis, Studies in Logic and the Foundations of Mathematics (North-Holland, Amsterdam, 1966).Google Scholar
[14]Schmid, J.. ‘Completing Boolean Algebras by Nonstandard Methods’, Zeitschr. f. Math. Logik und Grundlagen d. Math. 20 (1974), 4748.CrossRefGoogle Scholar
[15]Schmid, J., ‘Nonstandard Constructions for Join-extensions of Lattices’, Houston J. Math. 3 (3) (1977), 423439.Google Scholar
[16]Schweigert, D., ‘Central relations on lattices’, J. Austral. Math. Soc. (Series A) 35 (1983), 369372.CrossRefGoogle Scholar
[17]Schweigert, D., ‘Tolerances and Commutators on lattices’, Bull. Austral. Math. Soc. 37 (1988), 213219.CrossRefGoogle Scholar
[18]Schweigert, D. and Szymanska, M., ‘On central relations of complete lattices’, Czech. Math. J. 37 (1987), 7074.CrossRefGoogle Scholar