Definition:Associative Operation

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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.


Sources