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

Abstract

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“.