Two-dimensional Einstein numbers and associativity

TL;DR

This paper explores generalizations of Einstein numbers to higher-dimensional and abstract spaces, focusing on properties like associativity and commutativity of hyperbolic addition, and introduces two new two-dimensional Einstein number systems.

Contribution

It introduces two novel two-dimensional Einstein number systems and analyzes their algebraic properties, extending the theory to Hilbert-like and other abstract spaces.

Findings

Hyperbolic addition is associative and commutative in generalized settings.

Two new two-dimensional Einstein number systems are proposed.

Properties like distributivity are examined for the new operations.

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.