Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T00:54:44.433Z Has data issue: false hasContentIssue false
  • English
  • Français

Short Formulations of Boolean Algebra,Using Ring Operations

Published online by Cambridge University Press:  20 November 2018

Lee Byrne
Lee Byrne*
Affiliation:
Arizona State College
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.

Special interest has recently attached to formulations of Boolean algebra in terms of ring operations [7], [1]. These axiomatizations have not been as brief as those reached through other modes of approach.

The present note will show that the number of axioms when ring operations are used may be as small as in any present version that is not metamathematical, that is, the number of axioms finally employed will be two.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 31 - 33
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Bernstein, B. A., Postulates for Boolean algebra involving the operation of complete disjunction, Ann. of Math., vol. 37 (1936), 317–325; with references.Google Scholar
[2] Bernstein, B. A., Postulate sets for Boolean rings, Trans. Amer. Math. Soc, vol. 55 (1944), 393–400; with references.Google Scholar
[3] Birkhoff, Garrett, Lattice theory, Amer. Math. Soc. Colloquium Publ., vol. XXV (1940), chapt. VI.Google Scholar
[4] Byrne, Lee, Two brief formulations of Boolean algebra, Bull. Amer. Math Soc, vol. 52 (1946), 269–272, references.Google Scholar
[5] Byrne, Lee, Boolean algebra in terms of inclusion,Amer. J. Math., vol. 70 (1948), 139–143.Google Scholar
[6] Hoberman, S. and J. McKinsey, C. C., A set of postulates for Boolean algebra, Bull. Amer. Math. Soc, vol. 43 (1937), 588–592.Google Scholar
[7] Stone, M. H., Subsumption of the theory of Boolean algebras under the theory of rings, Proc Nat. Acad. Sci., U.S.A., vol. 21 (1935), 103–105.Google Scholar
[8] Stone, M. H., The theory of representations for Boolean algebras, Trans. Amer. Math. Soc, vol. 40 (1936), 37–111.Google Scholar