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Homomorphisms for distributive operations in partial algebras. Applications to linear operators and measure theory

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Summary

In this paper the concept of distributivity introduced earlier [1] is used to show that a homomorphism with respect to two distributive operations which is extended as a homomorphism with respect to operation 1 remains necessarily also a homomorphism with respect to operation 2 on the 1-closure of the original domain of definition. The result is illustrated by applications to continuous extensions of homomorphisms between δ-complete vector lattices, association of families of stochastically independent systems of sets and integration of products of independent functions.

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References

  1. [1]

    Baumann, V., andJ. Pfanzagl: The closure operator in partial algebras with distributive operations. Applications to set algebra, measure theory and linear spaces. Math. Zeitschr.92, 416–424 (1966).

  2. [2]

    Birkhoff, G.: Lattice theory. New York: Amer. Math. Soc. Coll. Publ. 1948.

  3. [3]

    Bourbaki, N.: Livre VI. Intégration, Chap. I–IV. Paris: Hermann 1965.

  4. [4]

    Halmos, P.R.: Measure theory. Princeton, N.J.: Van Nostrand 1950.

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Pfanzagl, J. Homomorphisms for distributive operations in partial algebras. Applications to linear operators and measure theory. Math Z 99, 270–278 (1967). https://doi.org/10.1007/BF01112456

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Keywords

  • Linear Operator
  • Vector Lattice
  • Measure Theory
  • Continuous Extension
  • Independent Function