diassociativity

Diassociativity is a property of binary operations in abstract algebra. A binary operation * on a set S is said to be diassociative if for all elements a, b, and c in S, the expressions (a * b) * c and a * (b * c) are not necessarily equal, but they are equal to some element d, which is uniquely determined by a, b, and c. This means that as long as you only use a fixed sequence of elements and the operation, the result is unambiguous, even if the order of operations is changed.

More formally, a magma (S, *) is diassociative if for any a, b, c in S, there exists a unique element d in S such that (a * b) * c = d and a * (b * c) = d. This unique element d is sometimes denoted as a <binary data, 1 bytes><binary data, 1 bytes> b <binary data, 1 bytes><binary data, 1 bytes> c, where the middle element b is the "pivot" or "center" of the diassociative product.

A common example of a diassociative operation is the standard multiplication of real numbers. While (2 * 3) * 4 = 6 * 4 = 24 and 2 * (3 * 4) = 2 * 12 = 24, and these are equal, diassociativity is a weaker condition than associativity. Diassociativity only guarantees that the two specific groupings, (a * b) * c and a * (b * c), yield the same result for any a, b, and c. It does not imply that other groupings, such as ((a * b) * c) * d, would also be equal to the same result if the operation were rearranged.

Diassociativity is a key property in the study of certain algebraic structures, particularly those that arise in fields like theoretical physics and in the analysis of non-associative algebras. It provides a way to handle computations with a binary operation where strict associativity might not hold, but a limited form of predictable grouping still exists.