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Postulates For Distributive Lattices

Published online by Cambridge University Press:  20 November 2018

Marlow Sholander
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Many sets of postulates have been given for distributive lattices and for Boolean algebra. For a description of some of the most interesting and for references to others the reader is referred to Birkhoff's “Lattice Theory”[1]. In this paper we give sets of postulates which have some intrinsic interest because of their simplicity. In the first two sections binary operations are used to describe a distributive lattice by 2 identities in 3 variables and a Boolean algebra by 3 identities in 3 variables. In the third section a ternary operation is used to describe distributive lattices with 0 and J by 2 identities in 5 variables.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 28 - 30
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Birkhoff, Garrett, Lattice Theory, Amer. Math. Soc. Colloquium Publications, vol. XXV' 1948.Google Scholar
[2] Croisot, R., Axiomatique des lattices distributives, Can. J. Math., vol. 3, (1951), pp. 24–27.Google Scholar
[3] Grau, A. A., Ternary operations and Boolean algebra, Ph.D. Thesis, University of Michigan, 1944.Google Scholar
[4] Kiss, S. A. and Birkhoff, Garrett, A ternary operation in distributive lattices, Bull. Amer. Math. Soc, vol. 53 (1947), 749–752.Google Scholar