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Boolean Averages

Published online by Cambridge University Press:  20 November 2018

Fred B. Wright
Fred B. Wright*
Affiliation:
The Tulane University of Louisiana
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The purpose of this note is to investigate the properties of a mapping α, of a Boolean algebra A into itself, which satisfies the functional equation α(p·aq) = ap·aq, where the multiplication is infimum. This is the so-called averaging identity of Kampé de Fériét. Garrett Birkhoff has noted (2) the occurrence of this identity both in the statistical theory of turbulence and in mathematical logic. That this is more than coincidence is shown by the existence of an explicit connection between the notion of quantification in logic and certain non-linear "extremal operators" in function algebras (16). Certain linear operators are associated in a natural way with these extremal operators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 440 - 455
Copyright
Copyright © Canadian Mathematical Society 1963

References

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