On one space of four-dimensional numbers

Abstract

The theory of four-dimensional numbers was first introduced by W.R. Hamilton in 1843, which is called the theory of quaternions. In this theory multiplication is a non-commutative operation, as a result of which it was not possible to construct a full-fledged mathematical analysis of functional spaces. In 2003, a new theory of functions of four-dimensional variables was published by Kazakh mathematicians B. Maukeev and M.M. Abenov, where commutative multiplication is introduced, which allows solving three-dimensional models of mechanics by the analytical method. A more complete presentation of the new theory was published in 2019 by M.M. Abenov in a monograph. While developing this theory, Abenov and M.B. Gabbasov found all four-dimensional spaces with commutative multiplication, which were assigned the designations M2 - M7, and it became necessary to study these spaces. This paper studies one of these spaces, namely the space of four-dimensional numbers M5. The purpose of this paper is to study the properties of four-dimensional numbers of the space M5 and substantiate its significance. In this study, new results are obtained on an algebra of the space M5, various norms and metrics are introduced, and the properties of number sequences are considered.