Associative binary operation

This article defines a property of binary operations (and hence, of magmas)

Definition

Definition with symbols

Let $S$ be a set and $*$ be a binary operation on $S$ (viz, $*$ is a map $S\times S\to S$). Then, $*$ is said to be associative if, for every $a,b,c$ in $S$, the following identity holds:

$(a*b)*c=a*(b*c)$

The expression on the left side is termed the left associated expression and the expression on the right side is termed the right associated expression. If, for a given $a,b,c$, the left associated expression and the right associated expression are equal, $a,b,c$ are said to associate. Associativity basically says that any ordered triple of elements associates.

Related term

A set equipped with an associative binary operation is termed a semigroup. If, further, there is a neutral element (identity element) for the associative binary operation, the set is termed a monoid.

Facts

Parenthesization can be dropped

When a binary operation is associative, it turns out that we can drop parenthesization from products of many elements. That is, given an expression of the form:

$a_{1}*a_{2}...*a_{n}$

any choice of bracketing will give the same result.

The result is proved by induction, with the base case ($n=3$) following from the definition of associativity.

For this reason, we always use infix operator symbols for associative binary operations, and often even drop the operator symbol, so that the above expression is just written as: $a_{1}a_{2}...a_{n}$.

Inverses are unique

In a monoid (that is, a set with associative binary operation having a neutral element) any left inverse and right inverse of an element must be equal. Hence, the inverse of an element, if it exists, must be unique. For full proof, refer: Inverse element#Equality of left and right inverse

Related element properties

Left associative element

An element is said to be left associative with respect to a binary operation if any ordered triple starting with that element associates.

The set of left associative elements in any magma is a subsemigroup, and if the magma contains a neutral element, it is a submonoid.

Middle associative element

An element is said to be 'middle associative with respect to a binary operation if any ordered triple with that element in the middle, associates.

The set of middle associative elements in any magma is a subsemigroup, and if the magma contains a neutral element, it is a submonoid.

Right associative element

An element is said to be right associative with respect to a binary operation if any ordered triple ending with that element associates.

The set of right associative elements in any magma is a subsemigroup, and if the magma contains a neutral element, it is a submonoid.

Associative element

Further information: associative element An element is said to be associative if it is left, middle and right associative. The set of associative elements forms a submagma (which contains the neutral element if it exists) termed the associative center of the magma.