Two-dimensional Einstein numbers and associativity

Abstract

In this paper, we deal with generalizations of real Einstein numbers to various spaces and dimensions. We search operations and their properties in generalized settings. Especially, we are interested in the generalized operation of hyperbolic addition to more-dimensional spaces, which is associative and commutative. We extend the theory to some abstract spaces, especially to Hilbert-like ones. Further, we bring two different two-dimensional generalizations of Einstein numbers and study properties of new-defined operations -- mainly associativity, commutativity, and distributive laws.