Associativity¶

Definition¶

A binary operation $\star$ on $A$ is associative if

(a\star b)\star c=a\star(b\star c)

for all $a,b,c\in A$ $A$ .

Interpretation¶

Associativity says that in a chain $a\star b\star c$ it does not matter which pair is combined first. Repeating this rule shows that every way of placing brackets in $a_1\star a_2\star\cdots\star a_n$ gives the same element, so expressions such as $a+b+c+d$ or $a\star a\star a$ have a meaning without brackets.

Associativity is about grouping, not about order: the elements keep their places, unlike in commutativity. An operation can be associative without being commutative. Composition of functions is associative, but for $f(x)=x+1$ and $g(x)=2x$ one has $(g\circ f)(x)=2x+2$ and $(f\circ g)(x)=2x+1$ $1$ . Subtraction is not associative: $(5-3)-2=0$ $0$ , but $5-(3-2)=4$ $4$ .

Associativity is the first requirement in the chain semigroup, monoid, group. It returns later with the product of matrices, which is associative but not commutative.

Key Results¶

Associativity is required in semigroups, monoids, and groups.
Associativity, identity elements, and inverse elements determine which algebraic structure is present.

Examples¶

Addition and multiplication of real numbers are associative. Subtraction is not associative.

Prerequisites¶

Binary Operation

Sources¶

Math II: Linear Algebra · Week 1

Aliases: associative property, associative law Symbols: element of

Last updated: 20 October 2026 03:02