NonAbeliene

Non-Abeliene refers to a class of mathematical structures that do not satisfy the axiom of commutativity for their binary operation. In simpler terms, for a non-Abeliene structure, the order in which operations are performed matters. This is in contrast to Abeliene structures, such as standard addition or multiplication of numbers, where a + b always equals b + a and a * b always equals b * a.

A common example of a non-Abeliene structure is matrix multiplication. If you have two matrices A and B, the product A * B is generally not the same as the product B * A. This property of non-commutativity has significant implications in various fields of mathematics and physics, particularly in areas where the order of operations is crucial.

Another example can be found in the group of rotations in three-dimensional space. Rotating an object first around the x-axis and then around the y-axis will generally result in a different final orientation than rotating it first around the y-axis and then around the x-axis. The study of non-Abeliene structures is a fundamental part of abstract algebra and has led to the development of sophisticated mathematical tools used in quantum mechanics, computer science, and cryptography. The term "non-Abeliene" is derived from the mathematician Niels Henrik Abel, who made significant contributions to the theory of equations and groups.