Non-associative

In mathematics and computer science, 'non-associative' describes an operation where the order of grouping elements using parentheses affects the final result. Specifically, if an operation * is non-associative, then for elements a, b, and c, (a * b) * c ≠ a * (b * c). This contrasts with associative operations, such as addition and multiplication of real numbers, where grouping does not alter the outcome. non-associative structures are often found in abstract algebra and are fundamental in defining more complex algebraic systems. These systems exhibit unique behaviors and properties not found in more common, associative operations.

Non-associative meaning with examples

• Consider octonion multiplication. This operation is non-associative, so the order of operations matters greatly. For example, (x * y) * z is often not equal to x * (y * z) for octonions x, y, and z. This lack of associativity gives rise to the octonions' interesting, albeit less intuitive, algebraic structure. It contrasts starkly with simpler number systems.
• Matrix multiplication, when dealing with more than two matrices, is associative, provided that the dimensions are compatible for the multiplication. Though it may seem that matrix multiplication could be considered non-associative based on the sheer complexity of the process and the order of matrix multiplication, this is simply not the case, and it adheres to the standard definition of associativity.
• The vector cross product in three dimensions is an example of a non-associative operation. The identity (a x b) x c ≠ a x (b x c) typically holds true, emphasizing how the parenthesization changes the end result. This property is critical in its applications in physics and geometry. In a related way, other vector spaces can be non-associative.
• In programming, consider a custom data structure with a defined operator. If that operator is implemented as a non-associative operation, the programmer must carefully manage the order of operations to prevent unintended consequences or incorrect results. The compiler will interpret non-associative operation based on the language specifications.
• Many types of algebras, such as Lie algebras, incorporate non-associative operations. Lie algebras are used extensively in representing symmetries in physics and other fields. They are typically not as intuitive as associative algebra systems, meaning they demand careful adherence to the defined order of operations and mathematical rules.