The Colombeau Quaternion Algebra

TL;DR

This paper introduces the Colombeau Quaternion Algebra, explores its algebraic properties, and establishes criteria for generalized holomorphic functions to satisfy the identity theorem, advancing the understanding of generalized function algebras.

Contribution

It presents the first study of the Colombeau Quaternion Algebra and analyzes its structure, including dense ideals and maximal quotients, with applications to generalized holomorphic functions.

Findings

Existence of a maximal quotient that is Von Neumann regular.

Characterization of dense ideals in the algebra of Colombeau generalized numbers.

Criteria for generalized holomorphic functions to satisfy the identity theorem.

Abstract

We introduce the Colombeau Quaternio Algebra and study its algebraic structure. We also study the dense ideal, dense in the algebraic sense, of the algebra of Colombeau generalized numbers and use this show the existence of a maximal ting of quotions which is Von Neumann regular. Recall that it is already known that then algebra of COlombeau generalized numbers is not Von Neumann regular. We also use the study of the dense ideals to give a criteria for a generalized holomorphic function to satisfy the identity theorem. Aragona-Fernadez-Juriaans showed that a generalized holomorphic function has a power series. This is one of the ingredients use to prove the identity theorem.