Conpseudosimilarity and consemisimilarity over a division ring

Solving Two Pairs of Quadratic Equations in Quaternions

“…Two quaternions a and b are said to be consemisimilar if there exist quaternions x, y such that (1.5) holds. The concept of con-semi-similarity was first introduced in [1,2]. In this section, we shall determine necessary and sufficient conditions such that (1.5) is consistent and give general common solutions to (1.5).…”

“…This is one of the challenging topics in the theory of quaternions. For the simplest equation ax = b in quaternions, where a and b are given and a = 0, its solution is x = a −1 b, where a −1 = (a 2 0 + a 2 1 + a 2 2 + a 2 3 ) −1 (a 0 − a 1 i − a 2 j − a 3 k) is the inverse of the nonzero quaternion a = a 0 + a 1 i + a 2 j + a 3 k. However, two-sided equations like ax + xb = c and axb+cxd = e in quaternions cannot be solved by any elementary methods, because 186 Y. Tian AACA the non-commutativity of quaternion multiplication makes it difficult to simplify the equations any further.…”

“…The semi-similarity was first introduced and studied by Hartwig and Putcha [3] for elements in a semigroup with identity 1. If there exist x and y that satisfy (1.5), a and b are said to be con-semi-similar, this concept was introduced by Bevis, Hall and Hartwig [1,2]. Both semi-similarity and con-semi-similarity are known as equivalence relations.…”