Definition:Associative Operation

Definition

Let $S$ be a set.

Let $\circ : S \times S \to S$ be a binary operation.

Then $\circ$ is associative if and only if:

$\forall x, y, z \in S: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$

Examples

$x \circ a \circ y$ Operation

Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is an associative operation.

Let $a \in S$ be an arbitrary element of $S$.

Let $*$ be the operation defined on $S$ by:

$\forall x, y \in S: x * y := x \circ a \circ y$

Then $*$ is associative on $S$.

Arbitrary Non-Associative Order 3 Structure

Consider the algebraic structure of order $3$ defined by the Cayley table:

$\begin{array}{c|cccc} \circ & a & b & c \\ \hline a & b & c & b \\ b & b & a & c \\ c & a & c & c \\ \end{array}$

 $\ds \paren {a \circ a} \circ b$ $=$ $\ds b \circ b$ $\ds$ $=$ $\ds a$ $\ds a \circ \paren {a \circ b}$ $=$ $\ds a \circ c$ $\ds$ $=$ $\ds b$

demonstrating non-associativity.

Also note that $a \circ b \ne b \circ a$, so $\circ$ is non-commutative as well.

$x y + 1$ Operation on Reals

Let $\R$ denote the set of real numbers.

Let $\circ$ denote the operation on $\R$ defined as:

$\forall x, y \in \R: x \circ y := x y + 1$

Then $\circ$ is not an associative operation, despite being commutative.

Also see

• Results about associativity can be found here.

Historical Note

The term associative was coined by William Hamilton in about $1844$ while thinking about octonions, which aren't.