Nonassociatiivisuus

Nonassociatiivisuus refers to a property of a binary operation in mathematics where the order of operations matters. Specifically, for an operation denoted by $\circ$, it is nonassociative if there exist elements $a$, $b$, and $c$ in the set over which the operation is defined such that $(a \circ b) \circ c \neq a \circ (b \circ c)$. This is in contrast to associative operations, where the grouping of operands does not affect the result.

Many common mathematical operations are associative. For example, addition and multiplication of real numbers are associative: $(a+b)+c = a+(b+c)$ and $(a \times b) \times c = a \times (b \times c)$. However, there are numerous important nonassociative operations.

Examples of nonassociative operations include subtraction, where $(5-3)-1 = 2-1 = 1$, but $5-(3-1) = 5-2 = 3$. Another classic example is the cross product of vectors in three-dimensional space. For vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$, it is generally true that $(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} \neq \mathbf{u} \times (\mathbf{v} \times \mathbf{w})$. Nonassociativity also appears in more abstract algebraic structures such as Lie algebras and Jordan algebras, which are specifically constructed to study operations with particular nonassociative properties. The study of nonassociative algebras is a significant area within abstract algebra.