Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.     

Isosceles Orthogonal Triples in Linear 2-Normed Spaces

A triple (x, y, z) in a linear 2-normed space (X, ‖.,.‖) is called an isosceles orthogonal triple, denoted |(x, y, z), if |(.,.,.) is said to be homogeneous if |(x, y, z) implies |(ax, y, z) for all real a and it is additive if |(x₁, y, z) and |(x₂, y, z) imply that |(x₁ + x₂, y, z). In addition to developing some basic properties of |(.,.,.), this paper shows that under the assumption of strict convexity, every subspace of X of dimension ≤ 3 contains an isosceles orthogonal triple. Further, if (X, ‖.,.‖) is strictly convex and |(…,.) is either homogeneous or additive, then (X, ‖.,.‖) is a 2-inner product space.     

Congruence modularity implies the Arguesian identity

It is shown that if is any variety of algebras all of whose congruence lattices are modular, then the congruence lattice of every algebra in satisfies the Arguesian law.     

Subject Index

Subject Index

Subject Index     

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.