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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

1. Birkhoff, Garrett, Moyennes des fonctions bornées, Colloque International, No. 24, CNRS, Paris, 1950.Google Scholar
2. Birkhoff, Garrett, Lattices in applied mathematics, Proc. Symposia Pure Math., Vol. II, pp. 155–184 (Providence, 1961).Google Scholar
3. Halmos, Paul R., Algebraic logic I, Comp. Math., 12 (1955), 217–259.Google Scholar
4. Henkin, Leon and Tarski, Alfred, Cylindric algebras, Proc. Symposia Pure Math., Vol. II, pp. 83–113 (Providence, 1961).Google Scholar
5. Jônsson, Bjarni and Tarski, Alfred, Boolean algebras with operators, Amer. J. Math., 73 (1951), 981–939.Google Scholar
6. Kelley, John L., Averaging operators on C(X), Illinois J. Math., 2 (1958), 214–223.Google Scholar
7. C, J. C.. McKinsey and Alfred Tarski, The algebra of topology, Ann. Math. (2), 45 (1944), 141–191.Google Scholar
8. Michael, Ernest, Topologies on spaces of subsets, Trans. Amer. Math. Soc, 71 (1951), 152–182.Google Scholar
9. Chen Moy, Shu-Teh, Characterization of conditional expectation as a transformation on function spaces, Pacific J. Math., 4 (1954), 47–63.Google Scholar
10. Pontrjagin, L., Topological groups (Princeton, 1946).Google Scholar
11. Rota, Gian-Carlo, On the representation of averaging operators, Rend. Sem. Mat. Padova, 30 (1960), 52–64.Google Scholar
12. Stone, Marshall H., The theory of representations for Boolean algebras, Trans. Amer. Math. Soc, 14(1937), 37–111.Google Scholar
13. Weil, André, Vintégration dans les groupes topologiques et ses applications (Paris, 1940).Google Scholar
14. Wright, Fred B., Some remarks on Boolean duality, Portugaliae Math., 16 (1957), 109–117.Google Scholar
15. Wright, Fred B., Recurrence theorems and operators on Boolean algebras, Proc. London Math. Soc. (3), 11 (1961), 385–401.Google Scholar
15. Wright, Fred B., Generalized means, Trans. Amer. Math. Soc, 98 (1961), 187–203.Google Scholar
17. Wright, Fred B., Convergence of quantifiers and martingales, Illinois J. Math., 6 (1962), 296–307.Google Scholar