Associative property

Formally, a binary operation ${\displaystyle \ast }$ on a set S is called associative if it satisfies the associative law:

${\displaystyle (x\ast y)\ast z=x\ast (y\ast z)}$, for all ${\displaystyle x,y,z}$ in S.

Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol (juxtaposition) as for multiplication.

${\displaystyle (xy)z=x(yz)}$, for all ${\displaystyle x,y,z}$ in S.

The associative law can also be expressed in functional notation thus: ${\displaystyle (f\circ (g\circ h))(x)=((f\circ g)\circ h)(x)}$

If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression.^[2] This is called the generalized associative law.

The number of possible bracketings is just the Catalan number, ${\displaystyle C_{n}}$ , for n operations on n+1 values. For instance, a product of 3 operations on 4 elements may be written (ignoring permutations of the arguments), in ${\displaystyle C_{3}=5}$ possible ways:

• ${\displaystyle ((ab)c)d}$
• ${\displaystyle (a(bc))d}$
• ${\displaystyle a((bc)d)}$
• ${\displaystyle a(b(cd))}$
• ${\displaystyle (ab)(cd)}$

If the product operation is associative, the generalized associative law says that all these expressions will yield the same result. So unless the expression with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as

${\displaystyle abcd}$

As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but they remain unnecessary for disambiguation.

An example where this does not work is the logical biconditional ↔. It is associative; thus, A ↔ (B ↔ C) is equivalent to (A ↔ B) ↔ C, but A ↔ B ↔ C most commonly means (A ↔ B) and (B ↔ C), which is not equivalent.

Some examples of associative operations include the following.

In standard truth-functional propositional logic, association,^[4][5] or associativity^[6] are two valid rules of replacement. The rules allow one to move parentheses in logical expressions in logical proofs. The rules (using logical connectives notation) are:

$${\displaystyle (P\lor (Q\lor R))\Leftrightarrow ((P\lor Q)\lor R)}$$

and

$${\displaystyle (P\land (Q\land R))\Leftrightarrow ((P\land Q)\land R),}$$

where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with".

Associativity is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that associativity is a property of particular connectives. The following (and their converses, since ↔ is commutative) are truth-functional tautologies.^{[citation needed]}

Associativity of disjunction

${\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}$

Associativity of conjunction

${\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}$

Associativity of equivalence

${\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}$

Joint denial is an example of a truth functional connective that is not associative.

A binary operation ${\displaystyle *}$ on a set S that does not satisfy the associative law is called non-associative. Symbolically,

$${\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{for some }}x,y,z\in S.}$$

For such an operation the order of evaluation does matter. For example:

Subtraction

${\displaystyle (5-3)-2\,\neq \,5-(3-2)}$

Division

${\displaystyle (4/2)/2\,\neq \,4/(2/2)}$

Exponentiation

${\displaystyle 2^{(1^{2})}\,\neq \,(2^{1})^{2}}$

Vector cross product

${\displaystyle {\begin{aligned}\mathbf {i} \times (\mathbf {i} \times \mathbf {j} )&=\mathbf {i} \times \mathbf {k} =-\mathbf {j} \\(\mathbf {i} \times \mathbf {i} )\times \mathbf {j} &=\mathbf {0} \times \mathbf {j} =\mathbf {0} \end{aligned}}}$

Also although addition is associative for finite sums, it is not associative inside infinite sums (series). For example, $${\displaystyle (1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots =0}$$ whereas $${\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots =1.}$$

Some non-associative operations are fundamental in mathematics. They appear often as the multiplication in structures called non-associative algebras, which have also an addition and a scalar multiplication. Examples are the octonions and Lie algebras. In Lie algebras, the multiplication satisfies Jacobi identity instead of the associative law; this allows abstracting the algebraic nature of infinitesimal transformations.

Other examples are quasigroup, quasifield, non-associative ring, and commutative non-associative magmas.

In mathematics, addition and multiplication of real numbers are associative. By contrast, in computer science, addition and multiplication of floating point numbers are not associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order.^[7]

To illustrate this, consider a floating-point representation with a 4-bit significand:

Even though most computers compute with 24 or 53 bits of significand,^[8] this is still an important source of rounding error, and approaches such as the Kahan summation algorithm are ways to minimize the errors. It can be especially problematic in parallel computing.^[9][10]

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like ${\displaystyle {\dfrac {2}{3/4}}}$). However, mathematicians agree on a particular order of evaluation for several common non-associative operations. This is simply a notational convention to avoid parentheses.

A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

$${\displaystyle \left.{\begin{array}{l}a*b*c=(a*b)*c\\a*b*c*d=((a*b)*c)*d\\a*b*c*d*e=(((a*b)*c)*d)*e\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}a,b,c,d,e\in S}$$

while a right-associative operation is conventionally evaluated from right to left:

$${\displaystyle \left.{\begin{array}{l}x*y*z=x*(y*z)\\w*x*y*z=w*(x*(y*z))\quad \\v*w*x*y*z=v*(w*(x*(y*z)))\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}z,y,x,w,v\in S}$$

Both left-associative and right-associative operations occur. Left-associative operations include the following:

Subtraction and division of real numbers^{[11][12][13][14][15]}

${\displaystyle x-y-z=(x-y)-z}$

${\displaystyle x/y/z=(x/y)/z}$

Function application

${\displaystyle (f\,x\,y)=((f\,x)\,y)}$

This notation can be motivated by the currying isomorphism, which enables partial application.

Right-associative operations include the following:

Exponentiation of real numbers in superscript notation

${\displaystyle x^{y^{z}}=x^{(y^{z})}}$

Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication:

${\displaystyle (x^{y})^{z}=x^{(yz)}}$

Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression ${\displaystyle 2^{x+3}}$ the addition is performed before the exponentiation despite there being no explicit parentheses ${\displaystyle 2^{(x+3)}}$ wrapped around it. Thus given an expression such as ${\displaystyle x^{y^{z}}}$, the full exponent ${\displaystyle y^{z}}$ of the base ${\displaystyle x}$ is evaluated first. However, in some contexts, especially in handwriting, the difference between ${\displaystyle {x^{y}}^{z}=(x^{y})^{z}}$, ${\displaystyle x^{yz}=x^{(yz)}}$ and ${\displaystyle x^{y^{z}}=x^{(y^{z})}}$ can be hard to see. In such a case, right-associativity is usually implied.

Function definition

${\displaystyle \mathbb {Z} \rightarrow \mathbb {Z} \rightarrow \mathbb {Z} =\mathbb {Z} \rightarrow (\mathbb {Z} \rightarrow \mathbb {Z} )}$

${\displaystyle x\mapsto y\mapsto x-y=x\mapsto (y\mapsto x-y)}$

Using right-associative notation for these operations can be motivated by the Curry–Howard correspondence and by the currying isomorphism.

Non-associative operations for which no conventional evaluation order is defined include the following.

Exponentiation of real numbers in infix notation^[16]

${\displaystyle (x^{\wedge }y)^{\wedge }z\neq x^{\wedge }(y^{\wedge }z)}$

Knuth's up-arrow operators

${\displaystyle a\uparrow \uparrow (b\uparrow \uparrow c)\neq (a\uparrow \uparrow b)\uparrow \uparrow c}$

${\displaystyle a\uparrow \uparrow \uparrow (b\uparrow \uparrow \uparrow c)\neq (a\uparrow \uparrow \uparrow b)\uparrow \uparrow \uparrow c}$

Taking the cross product of three vectors

${\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})\neq ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\qquad {\mbox{ for some }}{\vec {a}},{\vec {b}},{\vec {c}}\in \mathbb {R} ^{3}}$

Taking the pairwise average of real numbers

${\displaystyle {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\qquad {\mbox{for all }}x,y,z\in \mathbb {R} {\mbox{ with }}x\neq z.}$

Taking the relative complement of sets

${\displaystyle (A\backslash B)\backslash C\neq A\backslash (B\backslash C)}$.

(Compare material nonimplication in logic.)

William Rowan Hamilton seems to have coined the term "associative property"^[17] around 1844, a time when he was contemplating the non-associative algebra of the octonions he had learned about from John T. Graves.^[18]

In general, associative operations are not commutative. However, under certain special conditions, it may be the case that associativity implies commutativity. Associative operators defined on an interval of the real number line are commutative if they are continuous and injective in both arguments.^[19] A consequence is that every continuous, associative operator on two real inputs that is strictly increasing in each of its inputs is commutative.^[20]

Look up associative property in Wiktionary, the free dictionary.

• Light's associativity test
• Telescoping series, the use of addition associativity for cancelling terms in an infinite series
• A semigroup is a set with an associative binary operation.
• Commutativity and distributivity are two other frequently discussed properties of binary operations.
• Power associativity, alternativity, flexibility and N-ary associativity are weak forms of associativity.
• Moufang identities also provide a weak form of associativity.