# On quaternion valued rectangular S-metric space

### Abstract

A metric space can be seen as a distance space having a geometric structure, with only a few axioms. In this paper we introduce the concept of quaternion valued rectangular S metric spaces. The paper treats material concerning quaternion valued rectangular S metric spaces that is important for the study of fixed point theory in Clifford analysis. We introduce the basic ideas of quaternion valued rectangular S metric spaces and Cauchy sequences and discuss the completion of a quaternion valued rectangular S metric space. In this work, we will work on H, the skew field of quaternions. This means we can write each element q ∈ H in the form q = a + bi + cj + dk where a, b, c, d ∈ R and i, j, and k are the fundamental quaternion units. For these elements we have the multiplication rules I 2 = j 2 = k 2 = −1, ij = −ji = k, kj = −jk = −i and ki = −ik = j. The conjugate element is given by q = a−bi−cj −dk. The quaternion modulus has the form of |q|= √a 2 + b 2 + c 2 + d 2. Quaternions can be defined in several different equivalent ways. Quaternion is non commutative in multiplication. There is also more abstract possibilty of treating quaternions as simply quadruples of real numbers [a, b, c, d], with operation of addition and multiplication suitably defined. The components naturally group into the imaginary part (b, c, d), for which we take this part as a vector and the purely real part, a, which called a scalar. Sometimes, we write a quaternion as [a, V ] with V = (b, c, d). For more information about metric spaces, its generalization and quaternion analysis

*International Journal of Mathematical Sciences and Optimization: Theory and Applications*,

*10*(2), 58 - 70. Retrieved from http://ijmso.unilag.edu.ng/article/view/2080

Copyright (c) 2024 S. O. Ayodele , S. O. Ayodele , O. K. Adewale, B. E. Oyelade, O. F. Olayera, E. E. Aribike

This work is licensed under a Creative Commons Attribution 4.0 International License.

This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, adaptation, and reproduction in any medium, provided that the original work is properly cited.