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Autometrized Boolean Algebras I:Fundamental Distance-Theoretic Properties of B

Published online by Cambridge University Press:  20 November 2018

David Ellis
David Ellis*
Affiliation:
University of Missouri
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There have been several brief studies made [3, 4, 7, 8, 9, 11] of systems in which a “distance function” is defined on the set of pairs of elements of some abstract set to another abstract set. Frequently both of the sets involved are given algebraic structures. One of the more novel of these systems is the naturally metrized group [3, 7] originated by Karl Menger in 1931. This system is analogous to the Euclidean line in that it assigns to each pair, a, b of elements of an additively written Abelian group the “absolute value”, (a-b, b-a) = (b-a, a-b), of the "difference" of the elements as ”distance“.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 87 - 93
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Birkhoff, Garrett, Lattice Theory (revised edition), Amer. Math. Soc. Colloquium Publications, vol. XXV (1948).Google Scholar
[2] Blumenthal, L. M. and Ellis, David, Notes on metric lattices, Duke Math. J., vol. 16 (1949), 585–590.Google Scholar
[3] Ellis, David, Superposability properties of naturally metrized groups, Bull. Amer. Math. Soc, vol. 55 (1949), 639–640.Google Scholar
[4] Frechet, Maurice, De Vécart numérique à Vécart abstrait, Portugaliae Mathematica, vol. 5 (1946), 121–131.Google Scholar
[5] Glivenko, V., Géométrie des systémes de choses normées, Amer. J. Math., vol. 58 (1936),799–828.Google Scholar
[6] Glivenko, V., Contributions à Vétude des systémes de choses normées, Amer. J. Math., vol. 59 (1937), 941–956.Google Scholar
[7] Menger, Karl, Beitràge zur Gruppentheorie I. Über einen Abstand im Gruppen, Math. Z.f vol. 33 (1931), 396–418.Google Scholar
[8] Menger, Karl, Projective generalizations of metric geometry, Reports of a Mathema tical Colloquium (Notre Dame), Issue 5-6 (1944), 60–75.Google Scholar
[9] Menger, Karl, Statistical metrics, Proc. Nat. Acad. Sri., vol. 28 (1942), 535.Google Scholar
[10] Menger, Karl, Pitcher, Everett and Smiley, M. F., Transitivities of betweenness, Trans. Amer. Math. Soc, vol. 52 (1942), 95–114.Google Scholar
[11] Baley Price, G., A generalization of a metric space with applications to spaces whose elements are sets, Amer. J. Math., vol. 63 (1941), 546–560.Google Scholar