Associative property#Notation for non-associative operations

File:Semigroup_associative.svg. That is, when the two paths from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}} compose to the same function from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}}.]]

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

:$(x\; \backslash ast\; y)\; \backslash ast\; z\; =\; x\; \backslash ast\; (y\; \backslash ast\; z)$, for all $x,y,z$ in {{mvar|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.

:$(xy)z\; =\; x(yz)$, for all $x,y,z$ in {{mvar|S}}.

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

Image:Tamari lattice.svg of order four, possibly different products.]]

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.{{cite book |last=Durbin |first=John R. |title=Modern Algebra: an Introduction |year=1992 |publisher=Wiley |location=New York |isbn=978-0-471-51001-7 |page=78 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.html |edition=3rd |quote=If $a\_1,\; a\_2,\; \backslash dots,\; a\_n\; \backslash ,\backslash ,\; (n\; \backslash ge\; 2)$ are elements of a set with an associative operation, then the product $a\_1\; a\_2\; \backslash cdots\; a\_n$ is unambiguous; this is, the same element will be obtained regardless of how parentheses are inserted in the product.}} This is called the generalized associative law.

The number of possible bracketings is just the Catalan number, $C\_n$

, 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 $C\_3\; =\; 5$ possible ways:

• $((ab)c)d$
• $(a(bc))d$
• $a((bc)d)$
• $(a(b(cd))$
• $(ab)(cd)$

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

:$abcd$

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 {{math|↔}}. It is associative; thus, {{math|{{var|A}} ↔ ({{var|B}} ↔ {{var|C}})}} is equivalent to {{math|({{var|A}} ↔ {{var|B}}) ↔ {{var|C}}}}, but {{math|{{var|A}} ↔ {{var|B}} ↔ {{var|C}}}} most commonly means {{math|({{var|A}} ↔ {{var|B}}) and ({{var|B}} ↔ {{var|C}})}}, which is not equivalent.

File:Associativity of real number addition.svg

Some examples of associative operations include the following.

{{unordered list

|1= The concatenation of the three strings "hello", " ", "world" can be computed by concatenating the first two strings (giving "hello ") and appending the third string ("world"), or by joining the second and third string (giving " world") and concatenating the first string ("hello") with the result. The two methods produce the same result; string concatenation is associative (but not commutative).

|2= In arithmetic, addition and multiplication of real numbers are associative; i.e.,

\left.

\begin{matrix}

(x+y)+z=x+(y+z)=x+y+z\quad

\\

(x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \,

\end{matrix}

\right\}

\mbox{for all }x,y,z\in\mathbb{R}.

Because of associativity, the grouping parentheses can be omitted without ambiguity.

|3= The trivial operation {{math|1={{var|x}} ∗ {{var|y}} = {{var|x}}}} (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation $x\; \backslash circ\; y\; =\; y$ (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.

|4= Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative.

|5= The greatest common divisor and least common multiple functions act associatively.

\left.

\begin{matrix}

\operatorname{gcd}(\operatorname{gcd}(x,y),z)=

\operatorname{gcd}(x,\operatorname{gcd}(y,z))=

\operatorname{gcd}(x,y,z)\ \quad

\\

\operatorname{lcm}(\operatorname{lcm}(x,y),z)=

\operatorname{lcm}(x,\operatorname{lcm}(y,z))=

\operatorname{lcm}(x,y,z)\quad

\end{matrix}

\right\}\mbox{ for all }x,y,z\in\mathbb{Z}.

|6= Taking the intersection or the union of sets:

\left.

\begin{matrix}

(A\cap B)\cap C=A\cap(B\cap C)=A\cap B\cap C\quad

\\

(A\cup B)\cup C=A\cup(B\cup C)=A\cup B\cup C\quad

\end{matrix}

\right\}\mbox{for all sets }A,B,C.

|7= If {{mvar|M}} is some set and {{mvar|S}} denotes the set of all functions from {{mvar|M}} to {{mvar|M}}, then the operation of function composition on {{mvar|S}} is associative:$$(f\backslash circ\; g)\backslash circ\; h=f\backslash circ(g\backslash circ\; h)=f\backslash circ\; g\backslash circ\; h\backslash qquad\backslash mbox\{for\; all\; \}f,g,h\backslash in\; S.$$

|8= Slightly more generally, given four sets {{mvar|M}}, {{mvar|N}}, {{mvar|P}} and {{mvar|Q}}, with {{math|{{var|h}} : {{var|M}} → {{var|N}}}}, {{math|{{var|g}} : {{var|N}} → {{var|P}}}}, and {{math|{{var|f}} : {{var|P}} → {{var|Q}}}}, then

$$(f\backslash circ\; g)\backslash circ\; h=f\backslash circ(g\backslash circ\; h)=f\backslash circ\; g\backslash circ\; h$$

as before. In short, composition of maps is always associative.

|9= In category theory, composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms.

|10= Consider a set with three elements, {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}. The following operation:

{{wikitable| class="wikitable" style="text-align:center"

|-

! × !! {{mvar|A}} !! {{mvar|B}} !! {{mvar|C}}

|-

! {{mvar|A}}

| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}

|-

! {{mvar|B}}

| {{mvar|A}} || {{mvar|B}} || {{mvar|C}}

|-

! {{mvar|C}}

| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}

}}

is associative. Thus, for example, {{math|1={{var|A}}({{var|B}}{{var|C}}) = ({{var|A}}{{var|B}}){{var|C}} = {{var|A}}}}. This operation is not commutative.

|11= Because matrices represent linear functions, and matrix multiplication represents function composition, one can immediately conclude that matrix multiplication is associative.{{cite web|url=http://www.khanacademy.org/math/linear-algebra/matrix-transformations/composition-of-transformations/v/matrix-product-associativity|title=Matrix product associativity|publisher=Khan Academy|access-date=5 June 2016}}

|12= For real numbers (and for any totally ordered set), the minimum and maximum operation is associative: $$\backslash max(a,\; \backslash max(b,\; c))\; =\; \backslash max(\backslash max(a,\; b),\; c)\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash min(a,\; \backslash min(b,\; c))\; =\; \backslash min(\backslash min(a,\; b),\; c).$$

}}

{{Transformation rules}}

In standard truth-functional propositional logic, association,{{cite book |last1=Moore |first1=Brooke Noel |last2=Parker |first2=Richard |date=2017 |title=Critical Thinking |location=New York |publisher=McGraw-Hill Education |page=321 |isbn=9781259690877|edition=12th }}{{cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |last3=McMahon |first3=Kenneth |date=2014 |title=Introduction to Logic |location=Essex |publisher=Pearson Education |page=387 |isbn=9781292024820|edition=14th }} or associativity{{cite book |last1=Hurley |first1=Patrick J. |last2=Watson |first2=Lori |date=2016 |title=A Concise Introduction to Logic |location=Boston |publisher=Cengage Learning |page=427 |isbn=9781305958098|edition=13th }} 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:

$$(P\; \backslash lor\; (Q\; \backslash lor\; R))\; \backslash Leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash lor\; R)$$

and

$$(P\; \backslash land\; (Q\; \backslash land\; R))\; \backslash Leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash land\; R),$$

where "$\backslash Leftrightarrow$" 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 {{math|↔}} is commutative) are truth-functional tautologies.{{citation needed|reason=Stack Exchange is not a reliable source?|date=June 2022}}

;Associativity of disjunction

:$((P\; \backslash lor\; Q)\; \backslash lor\; R)\; \backslash leftrightarrow\; (P\; \backslash lor\; (Q\; \backslash lor\; R))$

;Associativity of conjunction

:$((P\; \backslash land\; Q)\; \backslash land\; R)\; \backslash leftrightarrow\; (P\; \backslash land\; (Q\; \backslash land\; R))$

;Associativity of equivalence

:$((P\; \backslash leftrightarrow\; Q)\; \backslash leftrightarrow\; R)\; \backslash leftrightarrow\; (P\; \backslash leftrightarrow\; (Q\; \backslash leftrightarrow\; R))$

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

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

$$(x*y)*z\backslash ne\; x*(y*z)\backslash qquad\backslash mbox\{for\; some\; \}x,y,z\backslash in\; S.$$

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

; Subtraction

:$$

(5-3)-2 \, \ne \, 5-(3-2)

; Division

:$$

(4/2)/2 \, \ne \, 4/(2/2)

; Exponentiation

:$$

2^{(1^2)} \, \ne \, (2^1)^2

; Vector cross product

:$\backslash begin\{align\}$

\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{align}

Also although addition is associative for finite sums, it is not associative inside infinite sums (series). For example,

(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots = 0

whereas

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.Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

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

{{block indent|1=(1.000₂×2⁰ + 1.000₂×2⁰) +

1.000₂×2⁴ = 1.000₂×2^{{{fontcolor|red|1}}} + 1.000₂×2⁴ = 1.00{{fontcolor|red|1}}₂×2⁴}}

{{block indent|1=1.000₂×2⁰ + (1.000₂×2⁰ +

1.000₂×2⁴) = 1.000₂×2^{{{fontcolor|red|0}}} + 1.000₂×2⁴ = 1.00{{fontcolor|red|0}}₂×2⁴}}

Even though most computers compute with 24 or 53 bits of significand,{{Cite book |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=29 August 2008 |id=IEEE Std 754-2008|doi=10.1109/IEEESTD.2008.4610935 |ref=CITEREFIEEE_7542008 |isbn=978-0-7381-5753-5}} this is still an important source of rounding error, and approaches such as the Kahan summation algorithm are ways to minimise the errors. It can be especially problematic in parallel computing.{{Citation

| last1 = Villa

| first1 = Oreste

| last2 = Chavarría-mir

| first2 = Daniel

| last3 = Gurumoorthi

| first3 = Vidhya

| last4 = Márquez

| first4 = Andrés

| last5 = Krishnamoorthy

| first5 = Sriram

| title = Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems

| url = http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf

| access-date = 8 April 2014

| archive-url = https://web.archive.org/web/20130215171724/http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf

| archive-date = 15 February 2013

| url-status = dead

}}{{cite journal|last=Goldberg|first=David|author-link=David Goldberg (PARC)|date=March 1991|title=What Every Computer Scientist Should Know About Floating-Point Arithmetic|url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|journal=ACM Computing Surveys|volume=23|issue=1|pages=5–48|doi=10.1145/103162.103163|s2cid=222008826|access-date=20 January 2016|url-status=live|archive-url=https://web.archive.org/web/20220519083509/http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|archive-date=2022-05-19}}

{{main|Operator associativity}}

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 $\backslash dfrac\{2\}\{3/4\}$). 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.,

\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:

\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 numbersGeorge Mark Bergman [https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html "Order of arithmetic operations"][http://eduplace.com/math/mathsteps/4/a/index.html "The Order of Operations"]. Education Place.[https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/v/introduction-to-order-of-operations "The Order of Operations"], timestamp [https://www.youtube.com/watch?v=ClYdw4d4OmA&t=5m40s 5m40s]. Khan Academy.[http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 "Using Order of Operations and Exploring Properties"] {{Webarchive|url=https://web.archive.org/web/20220716062834/http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 |date=2022-07-16 }}, section 9. Virginia Department of Education.Bronstein, :de:Taschenbuch der Mathematik, pages 115-120, chapter: 2.4.1.1, {{ISBN|978-3-8085-5673-3}}

:$x-y-z=(x-y)-z$

:$x/y/z=(x/y)/z$

; Function application

:$(f\; \backslash ,\; x\; \backslash ,\; y)\; =\; ((f\; \backslash ,\; x)\; \backslash ,\; y)$

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

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

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:

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

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

; Function definition

:$\backslash mathbb\{Z\}\; \backslash rarr\; \backslash mathbb\{Z\}\; \backslash rarr\; \backslash mathbb\{Z\}\; =\; \backslash mathbb\{Z\}\; \backslash rarr\; (\backslash mathbb\{Z\}\; \backslash rarr\; \backslash mathbb\{Z\})$

:$x\; \backslash mapsto\; y\; \backslash mapsto\; x\; -\; y\; =\; x\; \backslash mapsto\; (y\; \backslash mapsto\; x\; -\; y)$

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[https://codeplea.com/exponentiation-associativity-options Exponentiation Associativity and Standard Math Notation] Codeplea. 23 August 2016. Retrieved 20 September 2016.

:$(x^\backslash wedge\; y)^\backslash wedge\; z\backslash ne\; x^\backslash wedge(y^\backslash wedge\; z)$

; Knuth's up-arrow operators

:$a\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; c$

:$a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c$

; Taking the cross product of three vectors

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

; Taking the pairwise average of real numbers

:$\{(x+y)/2+z\backslash over2\}\backslash ne\{x+(y+z)/2\backslash over2\}\; \backslash qquad\; \backslash mbox\{for\; all\; \}x,y,z\backslash in\backslash mathbb\{R\}\; \backslash mbox\{\; with\; \}x\backslash ne\; z.$

; Taking the relative complement of sets

:$(A\backslash backslash\; B)\backslash backslash\; C\; \backslash neq\; A\backslash backslash\; (B\backslash backslash\; C)$.

(Compare material nonimplication in logic.)

William Rowan Hamilton seems to have coined the term "associative property"{{cite journal |author-link=William Rowan Hamilton |first=W.R. |last=Hamilton |year=1844–1850 |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ |title=On quaternions or a new system of imaginaries in algebra |journal=Philosophical Magazine |department=David R. Wilkins collection |publisher=Trinity College Dublin}} around 1844, a time when he was contemplating the non-associative algebra of the octonions he had learned about from John T. Graves.{{Cite journal | last1 = Baez | first1 = John C. | author-link = John Baez| title = The Octonions | journal = Bulletin of the American Mathematical Society | issn = 0273-0979 | volume = 39 | issue = 2 | pages = 145–205 | year = 2002 | url = https://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | doi = 10.1090/S0273-0979-01-00934-X | arxiv = math/0105155| mr = 1886087| s2cid = 586512}}

{{Wiktionary}}

• 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.

{{reflist}}

Category:Properties of binary operations

Category:Elementary algebra

Category:Functional analysis

Category:Rules of inference