Power associativityFrom Wikipedia, the free encyclopedia

Chapter 1

Absorbing element

In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation onthat set. The result of combining an absorbing element with any element of the set is the absorbing element itself.In semigroup theory, the absorbing element is called a zero element[1][2] because there is no risk of confusion withother notions of zero. In this article the two notions are synonymous. An absorbing element may also be called anannihilating element.

1.1 Definition

Formally, let (S, ∘) be a set S with a closed binary operation ∘ on it (known as a magma). A zero element is anelement z such that for all s in S, z∘s=s∘z=z. A refinement[2] are the notions of left zero, where one requires only thatz∘s=z, and right zero, where s∘z=z.Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring.In the case of a semiring with 0, the definition of an absorbing element is sometimes relaxed so that it is not requiredto absorb 0; otherwise, 0 would be the only absorbing element.[3]

1.2 Properties

• If a magma has both a left zero z and a right zero z′ , then it has a zero, since z = z × z′ = z′ .

• If a magma has a zero element, then the zero element is unique.

1.3 Examples

• The most well known example of an absorbing element in algebra is multiplication, where any number multi-plied by zero equals zero. Zero is thus an absorbing element.

• Floating point arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number(“NaN”). It is an absorbing element for every operation; i.e., x + NaN = NaN + x = NaN, x − NaN = NaN −x = NaN etc.

• The set of binary relations over a set X, together with the composition of relations forms a monoid with zero,where the zero element is the empty relation (empty set).

• The closed interval H=[0, 1] with x∘y=min(x,y) is also a monoid with zero, and the zero element is 0.

• More examples:

2

1.4. SEE ALSO 3

1.4 See also• Identity element

• Null semigroup

1.5 Notes[1] J.M. Howie, p. 2-3

[2] M. Kilp, U. Knauer, A.V. Mikhalev p. 14-15

[3] J.S. Golan p. 67

1.6 References• Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9.

• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Golan, Jonathan S. (1999). Semirings and Their Applications. Springer. ISBN 0-7923-5786-8.

1.7 External links• Absorbing element at PlanetMath

Chapter 2

Anticommutativity

Inmathematics, anticommutativity is the property of an operation with two ormore arguments wherein swapping theposition of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry,mathematical analysis and, as a consequence, in physics: they are often called antisymmetric operations.

2.1 Definition

An n -ary operation is anticommutative if swapping the order of any two arguments negates the result. For example,a binary operation ∗ is anti-commutative if for all x and y,

x ∗ y = −(y ∗ x).

More formally, a map ∗:An→G from the set of all n-tuples of elements in a set A (where n is a general integer) to agroup G is anticommutative if and only if

x1 ∗ x2 ∗ · · · ∗ xn = sgn(σ)(xσ(1) ∗ xσ(2) ∗ · · · ∗ xσ(n)) ∀x = (x1, x2, . . . , xn) ∈ An

where σ:(n)→(n) is an arbitrary permutation of the set (n) of the first n positive integers and sgn(σ) is its sign. Thisequality expresses the following concept:

• the value of the operation is unchanged, when applied to all ordered tuples constructed by even permutation ofthe elements of a fixed one.

• the value of the operation is the inverse of its value on a fixed tuple, when applied to all ordered tuples con-structed by odd permutation to the elements of the fixed one. The need for the existence of this inverse elementis the main reason for requiring the codomain G of the operation to be at least a group.

Note that this is an abuse of notation, since the codomain of the operation needs only to be a group: "−1” does nothave a precise meaning since a multiplication is not necessarily defined on G .Particularly important is the case n = 2. A binary operation ∗:A×A→G is anticommutative if and only if

x1 ∗ x2 = −(x2 ∗ x1) ∀(x1, x2) ∈ A×A

This means that x1 ∗ x2 is the inverse of the element x2 ∗ x1 in G .

2.2 Properties

If the group G is such that

4

2.3. EXAMPLES 5

−a = a ⇐⇒ a = 0 ∀a ∈ G

i.e. the only element equal to its inverse is the neutral element, then for all the ordered tuples such that xj = xi for atleast two different index i, j

x1 ∗ x2 ∗ · · · ∗ xn = 0

In the case n = 2 this means

x1 ∗ x1 = x2 ∗ x2 = 0

2.3 Examples

Examples of anticommutative binary operations include:

• Subtraction

• Cross product

• Lie bracket of a Lie algebra

• Lie bracket of a Lie ring

2.4 See also• Commutativity

• Commutator

• Exterior algebra

• Operation (mathematics)

• Symmetry in mathematics

• Particle statistics (for anticommutativity in physics).

2.5 References• Bourbaki, Nicolas (1989), “Chapter III. Tensor algebras, exterior algebras, symmetric algebras", Algebra.Chapters 1–3, Elements of Mathematics (2nd printing ed.), Berlin-Heidelberg-New York: Springer-Verlag,pp. xxiii+709, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001.

2.6 External links• Gainov, A.T. (2001), “Anti-commutative algebra”, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Anticommutative”, MathWorld.

Chapter 3

Associative property

This article is about associativity in mathematics. For associativity in the central processing unit memory cache, seeCPU cache. For associativity in programming languages, see operator associativity.“Associative” and “non-associative” redirect here. For associative and non-associative learning, see Learning#Types.

In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associa-tivity is a valid rule of replacement for expressions in logical proofs.Within an expression containing two or more occurrences in a row of the same associative operator, the order inwhich the operations are performed does not matter as long as the sequence of the operands is not changed. That is,rearranging the parentheses in such an expression will not change its value. Consider the following equations:

(2 + 3) + 4 = 2 + (3 + 4) = 9

2× (3× 4) = (2× 3)× 4 = 24.

Even though the parentheses were rearranged, the values of the expressions were not altered. Since this holds truewhen performing addition and multiplication on any real numbers, it can be said that “addition and multiplication ofreal numbers are associative operations”.Associativity is not to be confused with commutativity, which addresses whether a × b = b × a.Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups andcategories) explicitly require their binary operations to be associative.However, many important and interesting operations are non-associative; some examples include subtraction, exponentiationand the vector cross product. In contrast to the theoretical counterpart, the addition of floating point numbers in com-puter science is not associative, and is an important source of rounding error.

3.1 Definition

Formally, a binary operation ∗ on a set S is called associative if it satisfies the associative law:

(x ∗ y) ∗ z = x ∗ (y ∗ z) for all 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.

(xy)z = x(yz) = xyz for all x, y, z in S.

The associative law can also be expressed in functional notation thus: f(f(x, y), z) = f(x, f(y, z)).

6

3.2. GENERALIZED ASSOCIATIVE LAW 7

A binary operation ∗ on the set S is associative when this diagram commutes. That is, when the two paths from S×S×S to S composeto the same function from S×S×S to S.

3.2 Generalized associative law

If a binary operation is associative, repeated application of the operation produces the same result regardless how validpairs of parenthesis are inserted in the expression.[2] This is called the generalized associative law. For instance, aproduct of four elements may be written in five possible ways:

1. ((ab)c)d

2. (ab)(cd)

3. (a(bc))d

4. a((bc)d)

5. a(b(cd))

If the product operation is associative, the generalized associative law says that all these formulas will yield the sameresult, making the parenthesis unnecessary. Thus “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 theyremain unnecessary for disambiguation.

3.3 Examples

Some examples of associative operations include the following.

• The concatenation of the three strings “hello”, " ", “world” can be computed by concatenating the first twostrings (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 thesame result; string concatenation is associative (but not commutative).

• In arithmetic, addition and multiplication of real numbers are associative; i.e.,

8 CHAPTER 3. ASSOCIATIVE PROPERTY

(((ab)c)d)e

((ab)c)(de)

((ab)(cd))e

((a(bc))d)e

(ab)(c(de))

(a(bc))(de)

(ab)((cd)e)

(a(b(cd)))e

a(b(c(de)))

a((bc)(de))

a(b((cd)e))

a(((bc)d)e)

a((b(cd))e)

(a((bc)d))e

In the absence of the associative property, five factors a, b, c, d, e result in a Tamari lattice of order four, possibly different products.

(x+ y) + z = x+ (y + z) = x+ y + z(x y)z = x(y z) = x y z

}for all x, y, z ∈ R.

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

3.3. EXAMPLES 9

(x + z+ y)

x + z)+ (y

=

The addition of real numbers is associative.

• Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is alsoassociative, but multiplication of octonions is non-associative.

• The greatest common divisor and least common multiple functions act associatively.

gcd(gcd(x, y), z) = gcd(x, gcd(y, z)) = gcd(x, y, z)lcm(lcm(x, y), z) = lcm(x, lcm(y, z)) = lcm(x, y, z)

}for all x, y, z ∈ Z.

• Taking the intersection or the union of sets:

(A ∩B) ∩ C = A ∩ (B ∩ C) = A ∩B ∩ C(A ∪B) ∪ C = A ∪ (B ∪ C) = A ∪B ∪ C

}for all sets A,B,C.

• IfM is some set and S denotes the set of all functions fromM toM, then the operation of functional compositionon S is associative:

(f ◦ g) ◦ h = f ◦ (g ◦ h) = f ◦ g ◦ h for all f, g, h ∈ S.

• Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

(f ◦ g) ◦ h = f ◦ (g ◦ h) = f ◦ g ◦ h

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

• Consider a set with three elements, A, B, and C. The following operation:

is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative.

• Because matrices represent linear transformation functions, with matrix multiplication representing functionalcomposition, one can immediately conclude that matrix multiplication is associative.

10 CHAPTER 3. ASSOCIATIVE PROPERTY

3.4 Propositional logic

3.4.1 Rule of replacement

In standard truth-functional propositional logic, association,[3][4] or associativity[5] are two valid rules of replacement.The rules allow one to move parentheses in logical expressions in logical proofs. The rules are:

(P ∨ (Q ∨R)) ⇔ ((P ∨Q) ∨R)

and

(P ∧ (Q ∧R)) ⇔ ((P ∧Q) ∧R),

where "⇔ " is a metalogical symbol representing “can be replaced in a proof with.”

3.4.2 Truth functional connectives

Associativity is a property of some logical connectives of truth-functional propositional logic. The following logicalequivalences demonstrate that associativity is a property of particular connectives. The following are truth-functionaltautologies.Associativity of disjunction:

((P ∨Q) ∨R) ↔ (P ∨ (Q ∨R))

(P ∨ (Q ∨R)) ↔ ((P ∨Q) ∨R)

Associativity of conjunction:

((P ∧Q) ∧R) ↔ (P ∧ (Q ∧R))

(P ∧ (Q ∧R)) ↔ ((P ∧Q) ∧R)

Associativity of equivalence:

((P ↔ Q) ↔ R) ↔ (P ↔ (Q↔ R))

(P ↔ (Q↔ R)) ↔ ((P ↔ Q) ↔ R)

3.5 Non-associativity

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

(x ∗ y) ∗ z ̸= x ∗ (y ∗ z) for some x, y, z ∈ S.

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

• Subtraction

(5− 3)− 2 ̸= 5− (3− 2)

3.5. NON-ASSOCIATIVITY 11

• Division

(4/2)/2 ̸= 4/(2/2)

• Exponentiation

2(12) ̸= (21)2

Also note that infinite sums are not generally associative, for example:

(1− 1) + (1− 1) + (1− 1) + (1− 1) + (1− 1) + (1− 1) + . . . = 0

whereas

1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + (−1 + 1) + (−1 + 1) + (−1 + . . . = 1

The study of non-associative structures arises from reasons somewhat different from the mainstream of classicalalgebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associativelaw is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, andhave become ubiquitous in mathematics.There are other specific types of non-associative structures that have been studied in depth; these tend to come fromsome specific applications or areas such as combinatorial mathematics. Other examples are Quasigroup, Quasifield,Non-associative ring, Non-associative algebra and Commutative non-associative magmas.

3.5.1 Nonassociativity of floating point calculation

In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, theaddition and multiplication of floating point numbers is not associative, as rounding errors are introduced whendissimilar-sized values are joined together.[6]

To illustrate this, consider a floating point representation with a 4-bit mantissa:(1.0002×20 + 1.0002×20) + 1.0002×24 = 1.0002×21 + 1.0002×24 = 1.0012×241.0002×20 + (1.0002×20 + 1.0002×24) = 1.0002×20 + 1.0002×24 = 1.0002×24

Even though most computers compute with a 24 or 53 bits of mantissa,[7] this is an important source of roundingerror, and approaches such as the Kahan Summation Algorithm are ways to minimise the errors. It can be especiallyproblematic in parallel computing.[8][9]

3.5.2 Notation for non-associative operations

Main article: Operator associativity

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears morethan once in an expression. However, mathematicians agree on a particular order of evaluation for several commonnon-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.,

x ∗ y ∗ z = (x ∗ y) ∗ zw ∗ x ∗ y ∗ z = ((w ∗ x) ∗ y) ∗ zetc.

for all w, x, y, z ∈ S

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

12 CHAPTER 3. ASSOCIATIVE PROPERTY

x ∗ y ∗ z = x ∗ (y ∗ z)w ∗ x ∗ y ∗ z = w ∗ (x ∗ (y ∗ z))etc.

for all w, x, y, z ∈ S

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

• Subtraction and division of real numbers:

x− y − z = (x− y)− z for all x, y, z ∈ R;

x/y/z = (x/y)/z for all x, y, z ∈ R with y ̸= 0, z ̸= 0.

• Function application:

(f x y) = ((f x) y)

This notation can be motivated by the currying isomorphism.

Right-associative operations include the following:

• Exponentiation of real numbers:

xyz

= x(yz). .

The reason exponentiation is right-associative is that a repeated left-associative exponentiation operationwould be less useful. Multiple appearances could (and would) be rewritten with multiplication:

(xy)z = x(yz).

• Tetration via the up-arrow operator:

a ↑↑ b = ba

• Function definition

Z → Z → Z = Z → (Z → Z)

x 7→ y 7→ x− y = x 7→ (y 7→ x− y)

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

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

3.6. SEE ALSO 13

• Taking the Cross product of three vectors:

a⃗× (⃗b× c⃗) ̸= (⃗a× b⃗)× c⃗ for some a⃗, b⃗, c⃗ ∈ R3

• Taking the pairwise average of real numbers:

(x+ y)/2 + z

2̸= x+ (y + z)/2

2for all x, y, z ∈ R with x ̸= z.

• Taking the relative complement of sets (A\B)\C is not the same as A\(B\C) . (Compare material nonim-plication in logic.)

3.6 See also• Light’s associativity test

• A semigroup is a set with a closed associative binary operation.

• Commutativity and distributivity are two other frequently discussed properties of binary operations.

• Power associativity, alternativity and N-ary associativity are weak forms of associativity.

3.7 References[1] Thomas W. Hungerford (1974). Algebra (1st ed.). Springer. p. 24. ISBN 0387905189. Definition 1.1 (i) a(bc) = (ab)c

for all a, b, c in G.

[2] Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 0-471-51001-7. Ifa1, a2, . . . , an (n ≥ 2) are elements of a set with an associative operation, then the product a1a2 . . . an is unambiguous;this is, the same element will be obtained regardless of how parentheses are inserted in the product

[3] Moore and Parker

[4] Copi and Cohen

[5] Hurley

[6] Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

[7] IEEEComputer Society (August 29, 2008). “IEEE Standard for Floating-Point Arithmetic”. IEEE. doi:10.1109/IEEESTD.2008.4610935.ISBN 978-0-7381-5753-5. IEEE Std 754-2008.

[8] Villa, Oreste; Chavarría-mir, Daniel; Gurumoorthi, Vidhya; Márquez, Andrés; Krishnamoorthy, Sriram, Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems (PDF), retrieved 2014-04-08

[9] Goldberg, David (March 1991). “What Every Computer Scientist Should Know About Floating-Point Arithmetic” (PDF).ACM Computing Surveys 23 (1): 5–48. doi:10.1145/103162.103163. Retrieved 2016-01-20. (, )

Chapter 4

Commutative property

For other uses, see Commute (disambiguation).In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

=

==

This image illustrates that addition is commutative.

It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar asthe name of the property that says “3 + 4 = 4 + 3” or “2 × 5 = 5 × 2”, the property can also be used in more advancedsettings. The name is needed because there are operations, such as division and subtraction, that do not have it (forexample, “3 − 5 ≠ 5 − 3”), such operations are not commutative, or noncommutative operations. The idea that simpleoperations, such as multiplication and addition of numbers, are commutative was for many years implicitly assumedand the property was not named until the 19th century when mathematics started to become formalized.

4.1 Common uses

The commutative property (or commutative law) is a property generally associatedwith binary operations and functions.If the commutative property holds for a pair of elements under a certain binary operation then the two elements aresaid to commute under that operation.

14

4.2. MATHEMATICAL DEFINITIONS 15

4.2 Mathematical definitions

Further information: Symmetric function

The term “commutative” is used in several related senses.[1][2]

1. A binary operation ∗ on a set S is called commutative if:x ∗ y = y ∗ x for all x, y ∈ S

An operation that does not satisfy the above property is called non-commutative.2. One says that x commutes with y under ∗ if:x ∗ y = y ∗ x

3. A binary function f : A×A→ B is called commutative if:f(x, y) = f(y, x) for all x, y ∈ A

4.3 Examples

4.3.1 Commutative operations in everyday life• Putting on socks resembles a commutative operation since which sock is put on first is unimportant. Eitherway, the result (having both socks on), is the same. In contrast, putting on underwear and trousers is notcommutative.

• The commutativity of addition is observed when paying for an item with cash. Regardless of the order the billsare handed over in, they always give the same total.

4.3.2 Commutative operations in mathematics

Two well-known examples of commutative binary operations:[1]

• The addition of real numbers is commutative, since

y + z = z + y for all y, z ∈ R

For example 4 + 5 = 5 + 4, since both expressions equal 9.

• The multiplication of real numbers is commutative, since

yz = zy for all y, z ∈ R

For example, 3 × 5 = 5 × 3, since both expressions equal 15.

• Some binary truth functions are also commutative, since the truth tables for the functions are the same whenone changes the order of the operands.

For example, the logical biconditional function p↔ q is equivalent to q↔ p. This function is also writtenas p IFF q, or as p ≡ q, or as Epq.The last form is an example of the most concise notation in the article on truth functions, which lists thesixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp;Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

• Further examples of commutative binary operations include addition and multiplication of complex numbers,addition and scalar multiplication of vectors, and intersection and union of sets.

16 CHAPTER 4. COMMUTATIVE PROPERTY

b

b

aaa+

b

The addition of vectors is commutative, because a⃗+ b⃗ = b⃗+ a⃗ .

4.3.3 Noncommutative operations in everyday life

• Concatenation, the act of joining character strings together, is a noncommutative operation. For example,

EA+ T = EAT ̸= TEA = T + EA

• Washing and drying clothes resembles a noncommutative operation; washing and then drying produces amarkedly different result to drying and then washing.

• Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientationthan when the rotations are performed in the opposite order.

• The twists of the Rubik’s Cube are noncommutative. This can be studied using group theory.

• Also thought processes are noncommutative: A person asked a question (A) and then a question (B) may givedifferent answers to each question than a person asked first (B) and then (A), because asking a question maychange the person’s state of mind.

4.4. HISTORY AND ETYMOLOGY 17

4.3.4 Noncommutative operations in mathematics

Some noncommutative binary operations:[3]

• Subtraction is noncommutative, since 0− 1 ̸= 1− 0

• Division is noncommutative, since 1/2 ̸= 2/1

• Some truth functions are noncommutative, since the truth tables for the functions are different when one changesthe order of the operands.

For example, the truth tables for f (A,B) = A Λ ¬B (A AND NOT B) and f (B,A) = B Λ ¬A are

• Matrix multiplication is noncommutative since

[0 20 1

]=

[1 10 1

]·[0 10 1

]̸=

[0 10 1

]·[1 10 1

]=

[0 10 1

]• The vector product (or cross product) of two vectors in three dimensions is anti-commutative, i.e., b × a = −(a× b).

4.4 History and etymology

The first known use of the term was in a French Journal published in 1814

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commuta-tive property of multiplication to simplify computing products.[4][5] Euclid is known to have assumed the commutativeproperty of multiplication in his book Elements.[6] Formal uses of the commutative property arose in the late 18th andearly 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative propertyis a well known and basic property used in most branches of mathematics.The first recorded use of the term commutative was in a memoir by François Servois in 1814,[7][8] which used theword commutatives when describing functions that have what is now called the commutative property. The word is acombination of the French word commuter meaning “to substitute or switch” and the suffix -ative meaning “tendingto” so the word literally means “tending to substitute or switch.” The term then appeared in English in PhilosophicalTransactions of the Royal Society in 1844.[7]

18 CHAPTER 4. COMMUTATIVE PROPERTY

4.5 Propositional logic

4.5.1 Rule of replacement

In truth-functional propositional logic, commutation,[9][10] or commutativity[11] refer to two valid rules of replacement.The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

(P ∨Q) ⇔ (Q ∨ P )

and

(P ∧Q) ⇔ (Q ∧ P )

where "⇔ " is a metalogical symbol representing “can be replaced in a proof with.”

4.5.2 Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logicalequivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functionaltautologies.

Commutativity of conjunction (P ∧Q) ↔ (Q ∧ P )

Commutativity of disjunction (P ∨Q) ↔ (Q ∨ P )

Commutativity of implication (also called the law of permutation)(P → (Q→ R)) ↔ (Q→ (P → R))

Commutativity of equivalence (also called the complete commutative law of equivalence)(P ↔ Q) ↔ (Q↔ P )

4.6 Set theory

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the com-mutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity ofwell-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitlyassumed) in proofs.[12][13][14]

4.7 Mathematical structures and commutativity

• A commutative semigroup is a set endowed with a total, associative and commutative operation.

• If the operation additionally has an identity element, we have a commutative monoid

• An abelian group, or commutative group is a group whose group operation is commutative.[13]

• Acommutative ring is a ringwhosemultiplication is commutative. (Addition in a ring is always commutative.)[15]

• In a field both addition and multiplication are commutative.[16]

4.8. RELATED PROPERTIES 19

4.8 Related properties

4.8.1 Associativity

Main article: Associative property

The associative property is closely related to the commutative property. The associative property of an expressioncontaining two or more occurrences of the same operator states that the order operations are performed in does notaffect the final result, as long as the order of terms doesn't change. In contrast, the commutative property states thatthe order of the terms does not affect the final result.Most commutative operations encountered in practice are also associative. However, commutativity does not implyassociativity. A counterexample is the function

f(x, y) =x+ y

2,

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, forexample, f(−4, f(0,+4)) = −1 but f(f(−4, 0),+4) = +1 ). More such examples may be found in Commutativenon-associative magmas.

4.8.2 Symmetry

-10-8-6-4-2 0 2 4 6 8 10

Graph showing the symmetry of the addition function

Main article: Symmetry in mathematics

20 CHAPTER 4. COMMUTATIVE PROPERTY

Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as abinary function then the resulting function is symmetric across the line y = x. As an example, if we let a function frepresent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seenin the image on the right.For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, thenaRb⇔ bRa .

4.9 Non-commuting operators in quantum mechanics

Main article: Canonical commutation relation

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x(meaning multiply by x), and d

dx . These two operators do not commute as may be seen by considering the effect oftheir compositions x d

dx and ddxx (also called products of operators) on a one-dimensional wave function ψ(x) :

xd

dxψ = xψ′ ̸= d

dxxψ = ψ + xψ′

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do notcommute, then that pair of variables are mutually complementary, which means they cannot be simultaneously mea-sured or known precisely. For example, the position and the linear momentum in the x-direction of a particle arerepresented respectively by the operators x and −iℏ ∂

∂x (where ℏ is the reduced Planck constant). This is the sameexample except for the constant −iℏ , so again the operators do not commute and the physical meaning is that theposition and linear momentum in a given direction are complementary.

4.10 See also

• Anticommutativity

• Associative Property

• Binary operation

• Centralizer or Commutant

• Commutative diagram

• Commutative (neurophysiology)

• Commutator

• Distributivity

• Parallelogram law

• Particle statistics (for commutativity in physics)

• Quasi-commutative property

• Symmetric relation

• Trace monoid

• Truth function

• Truth table

4.11. NOTES 21

4.11 Notes

[1] Krowne, p.1

[2] Weisstein, Commute, p.1

[3] Yark, p.1

[4] Lumpkin, p.11

[5] Gay and Shute, p.?

[6] O'Conner and Robertson, Real Numbers

[7] Cabillón and Miller, Commutative and Distributive

[8] O'Conner and Robertson, Servois

[9] Moore and Parker

[10] Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

[11] Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

[12] Axler, p.2

[13] Gallian, p.34

[14] p. 26,87

[15] Gallian p.236

[16] Gallian p.250

4.12 References

4.12.1 Books

• Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2.

Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.

• Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

• Gallian, Joseph (2006). Contemporary Abstract Algebra, 6e. Boston, Mass.: Houghton Mifflin. ISBN 0-618-51471-6.

Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

• Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry, 2e. Prentice Hall. ISBN0-13-067342-0.

Abstract algebra theory. Uses commutativity property throughout book.

• Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

22 CHAPTER 4. COMMUTATIVE PROPERTY

4.12.2 Articles

• http://www.ethnomath.org/resources/lumpkin1997.pdf Lumpkin, B. (1997). The Mathematical Legacy OfAncient Egypt - A Response To Robert Palter. Unpublished manuscript.

Article describing the mathematical ability of ancient civilizations.

• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text.London: British Museum Publications Limited. ISBN 0-7141-0944-4

Translation and interpretation of the Rhind Mathematical Papyrus.

4.12.3 Online resources

• Hazewinkel, Michiel, ed. (2001), “Commutativity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Krowne, Aaron, Commutative at PlanetMath.org., Accessed 8 August 2007.

Definition of commutativity and examples of commutative operations

• Weisstein, Eric W., “Commute”, MathWorld., Accessed 8 August 2007.

Explanation of the term commute

• Yark. Examples of non-commutative operations at PlanetMath.org., Accessed 8 August 2007

Examples proving some noncommutative operations

• O'Conner, J J and Robertson, E F. MacTutor history of real numbers, Accessed 8 August 2007

Article giving the history of the real numbers

• Cabillón, Julio and Miller, Jeff. Earliest Known Uses Of Mathematical Terms, Accessed 22 November 2008

Page covering the earliest uses of mathematical terms

• O'Conner, J J and Robertson, E F. MacTutor biography of François Servois, Accessed 8 August 2007

Biography of Francois Servois, who first used the term

Chapter 5

Distributive property

“Distributivity” redirects here. It is not to be confused with Distributivism.

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive lawfrom elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rulesallow one to reformulate conjunctions and disjunctions within logical proofs.For example, in arithmetic:

2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).

In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the1 and the 3 individually, with the products added afterwards. Because these give the same final answer (8), it is saidthat multiplication by 2 distributes over addition of 1 and 3. Since one could have put any real numbers in place of2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes overaddition of real numbers.

5.1 Definition

Given a set S and two binary operators ∗ and + on S, we say that the operation:∗ is left-distributive over + if, given any elements x, y, and z of S,

x ∗ (y + z) = (x ∗ y) + (x ∗ z),

∗ is right-distributive over + if, given any elements x, y, and z of S,

(y + z) ∗ x = (y ∗ x) + (z ∗ x),

∗ is distributive over + if it is left- and right-distributive.[1] Notice that when ∗ is commutative, the three conditionsabove are logically equivalent.

5.2 Meaning

The operators used for examples in this section are the binary operations of addition ( + ) and multiplication ( · ) ofnumbers.There is a distinction between left-distributivity and right-distributivity:

23

24 CHAPTER 5. DISTRIBUTIVE PROPERTY

a · (b± c) = a · b± a · c (left-distributive)(a± b) · c = a · c± b · c (right-distributive)

In either case, the distributive property can be described in words as:To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factorand the resulting products are added (or subtracted).If the operation outside the parentheses (in this case, themultiplication) is commutative, then left-distributivity impliesright-distributivity and vice versa.One example of an operation that is “only” right-distributive is division, which is not commutative:

(a± b)÷ c = a÷ c± b÷ c

In this case, left-distributivity does not apply:

a÷ (b± c) ̸= a÷ b± a÷ c

The distributive laws are among the axioms for rings and fields. Examples of structures in which two operations aremutually related to each other by the distributive law are Boolean algebras such as the algebra of sets or the switchingalgebra. There are also combinations of operations that are not mutually distributive over each other; for example,addition is not distributive over multiplication.Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of asum with each summand of the other sums (keeping track of signs), and then adding up all of the resulting products.

5.3 Examples

5.3.1 Real numbers

In the following examples, the use of the distributive law on the set of real numbers R is illustrated. When multi-plication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point ofview of algebra, the real numbers form a field, which ensures the validity of the distributive law.

First example (mental and written multiplication)

During mental arithmetic, distributivity is often used unconsciously:

6 · 16 = 6 · (10 + 6) = 6 · 10 + 6 · 6 = 60 + 36 = 96

Thus, to calculate 6 ⋅ 16 in your head, you first multiply 6 ⋅ 10 and 6 ⋅ 6 and add the intermediate results. Writtenmultiplication is also based on the distributive law.

Second example (with variables)

3a2b · (4a− 5b) = 3a2b · 4a− 3a2b · 5b = 12a3b− 15a2b2

Third example (with two sums)

(a+ b) · (a− b) = a · (a− b) + b · (a− b) = a2 − ab+ ba− b2 = a2 − b2

= (a+ b) · a− (a+ b) · b = a2 + ba− ab− b2 = a2 − b2

Here the distributive law was applied twice and. It does not matter which bracket is first multiplied out.

5.4. PROPOSITIONAL LOGIC 25

Fourth Example Here the distributive law is applied the other way around compared to the previous examples.Consider12a3b2 − 30a4bc+ 18a2b3c2 .

Since the factor 6a2b occurs in all summand, it can be factored out. That is, due to the distributive law one obtains12a3b2 − 30a4bc+ 18a2b3c2 = 6a2b(2ab− 5a2c+ 3b2c2) .

5.3.2 Matrices

The distributive law is valid for matrix multiplication. More precisely,

(A+B) · C = A · C +B · C

for all l ×m -matrices A,B andm× n -matrices C , as well as

A · (B + C) = A ·B +A · C

for all l ×m -matrices A and m × n -matrices B,C . Because the commutative property does not hold for matrixmultiplication, the second law does not follow from the first law. In this case, they are two different laws.

5.3.3 Other examples1. Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.2. The cross product is left- and right-distributive over vector addition, though not commutative.3. The union of sets is distributive over intersection, and intersection is distributive over union.4. Logical disjunction (“or”) is distributive over logical conjunction (“and”), and conjunction is distributive over

disjunction.5. For real numbers (and for any totally ordered set), the maximum operation is distributive over the minimum

operation, and vice versa: max(a, min(b, c)) = min(max(a, b), max(a, c)) and min(a, max(b, c)) = max(min(a,b), min(a, c)).

6. For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(a,lcm(b, c)) = lcm(gcd(a, b), gcd(a, c)) and lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c)).

7. For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a+ max(b, c) = max(a + b, a + c) and a + min(b, c) = min(a + b, a + c).

5.4 Propositional logic

5.4.1 Rule of replacement

In standard truth-functional propositional logic, distribution[2][3][4] in logical proofs uses two valid rules of replacementto expand individual occurrences of certain logical connectives, within some formula, into separate applications ofthose connectives across subformulas of the given formula. The rules are:

(P ∧ (Q ∨R)) ⇔ ((P ∧Q) ∨ (P ∧R))

and

(P ∨ (Q ∧R)) ⇔ ((P ∨Q) ∧ (P ∨R))

where " ⇔ ", also written ≡, is a metalogical symbol representing “can be replaced in a proof with” or “is logicallyequivalent to”.

26 CHAPTER 5. DISTRIBUTIVE PROPERTY

5.4.2 Truth functional connectives

Distributivity is a property of some logical connectives of truth-functional propositional logic. The following logicalequivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functionaltautologies.

Distribution of conjunction over conjunction (P ∧ (Q ∧R)) ↔ ((P ∧Q) ∧ (P ∧R))

Distribution of conjunction over disjunction [5] (P ∧ (Q ∨R)) ↔ ((P ∧Q) ∨ (P ∧R))

Distribution of disjunction over conjunction [6] (P ∨ (Q ∧R)) ↔ ((P ∨Q) ∧ (P ∨R))

Distribution of disjunction over disjunction (P ∨ (Q ∨R)) ↔ ((P ∨Q) ∨ (P ∨R))

Distribution of implication (P → (Q→ R)) ↔ ((P → Q) → (P → R))

Distribution of implication over equivalence (P → (Q↔ R)) ↔ ((P → Q) ↔ (P → R))

Distribution of disjunction over equivalence (P ∨ (Q↔ R)) ↔ ((P ∨Q) ↔ (P ∨R))

Double distribution((P ∧Q) ∨ (R ∧ S)) ↔ (((P ∨R) ∧ (P ∨ S)) ∧ ((Q ∨R) ∧ (Q ∨ S)))((P ∨Q) ∧ (R ∨ S)) ↔ (((P ∧R) ∨ (P ∧ S)) ∨ ((Q ∧R) ∨ (Q ∧ S)))

5.5 Distributivity and rounding

In practice, the distributive property of multiplication (and division) over addition may appear to be compromised orlost because of the limitations of arithmetic precision. For example, the identity ⅓ + ⅓ + ⅓ = (1 + 1 + 1) / 3 appearsto fail if the addition is conducted in decimal arithmetic; however, if many significant digits are used, the calculationwill result in a closer approximation to the correct results. For example, if the arithmetical calculation takes the form:0.33333 + 0.33333 + 0.33333 = 0.99999 ≠ 1, this result is a closer approximation than if fewer significant digits hadbeen used. Even when fractional numbers can be represented exactly in arithmetical form, errors will be introducedif those arithmetical values are rounded or truncated. For example, buying two books, each priced at £14.99 beforea tax of 17.5%, in two separate transactions will actually save £0.01, over buying them together: £14.99 × 1.175= £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98 × 1.175 = £35.23. Methods suchas banker’s rounding may help in some cases, as may increasing the precision used, but ultimately some calculationerrors are inevitable.

5.6 Distributivity in rings

Distributivity is most commonly found in rings and distributive lattices.A ring has two binary operations (commonly called "+" and "∗"), and one of the requirements of a ring is that ∗ mustdistribute over +. Most kinds of numbers (example 1) and matrices (example 4) form rings. A lattice is another kindof algebraic structure with two binary operations, ∧ and ∨. If either of these operations (say ∧) distributes over theother (∨), then ∨ must also distribute over ∧, and the lattice is called distributive. See also the article on distributivity(order theory).Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring) ora special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributivelaws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.Failure of one of the two distributive laws brings about near-rings and near-fields instead of rings and division ringsrespectively. The operations are usually configured to have the near-ring or near-field distributive on the right but noton the left.Rings and distributive lattices are both special kinds of rigs, certain generalizations of rings. Those numbers inexample 1 that don't form rings at least form rigs. Near-rigs are a further generalization of rigs that are left-distributivebut not right-distributive; example 2 is a near-rig.

5.7. GENERALIZATIONS OF DISTRIBUTIVITY 27

5.7 Generalizations of distributivity

In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of theabove conditions or the extension to infinitary operations. Especially in order theory one finds numerous importantvariants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others beingdefined in the presence of only one binary operation, such as the according definitions and their relations are given inthe article distributivity (order theory). This also includes the notion of a completely distributive lattice.In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either ≤ or ≥.Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notionof sub-distributivity as explained in the article on interval arithmetic.In category theory, if (S, μ, η) and (S′, μ′, η′) are monads on a category C, a distributive law S.S′ → S′.S is a naturaltransformation λ : S.S′ → S′.S such that (S′, λ) is a lax map of monads S → S and (S, λ) is a colax map of monads S′→ S′. This is exactly the data needed to define a monad structure on S′.S: the multiplication map is S′μ.μ′S2.S′λS andthe unit map is η′S.η. See: distributive law between monads.A generalized distributive law has also been proposed in the area of information theory.

5.7.1 Notions of antidistributivity

The ubiquitous identity that relates inverses to the binary operation in any group, namely (xy)−1 = y−1x−1, whichis taken as an axiom in the more general context of a semigroup with involution, has sometimes been called anantidistributive property (of inversion as a unary operation).[7]

In the context of a near-ring, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements. Thelatter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements dis-tribute when multiplied on the left), then an antidistributive element a reverses the order of addition when multipliedto the right: (x + y)a = ya + xa.[8]

In the study of propositional logic and Boolean algebra, the term antidistributive law is sometimes used to denotethe interchange between conjunction and disjunction when implication factors over them:[9]

• (a ∨ b) ⇒ c ≡ (a⇒ c) ∧ (b⇒ c)

• (a ∧ b) ⇒ c ≡ (a⇒ c) ∨ (b⇒ c)

These two tautologies are a direct consequence of the duality in De Morgan’s laws.

5.8 Notes[1] Ayres, p. 20

[2] Moore and Parker

[3] Copi and Cohen

[4] Hurley

[5] Russell and Whitehead, Principia Mathematica

[6] Russell and Whitehead, Principia Mathematica

[7] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer. p. 4. ISBN978-3-211-82971-4.

[8] Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups.Kluwer Academic Publishers. pp. 62 and 67. ISBN 978-1-4613-0267-4.

[9] Eric C.R. Hehner (1993). A Practical Theory of Programming. Springer Science & Business Media. p. 230. ISBN978-1-4419-8596-5.

28 CHAPTER 5. DISTRIBUTIVE PROPERTY

5.9 References• Ayres, Frank, Schaum’s Outline of Modern Abstract Algebra, McGraw-Hill; 1st edition (June 1, 1965). ISBN0-07-002655-6.

5.10 External links• A demonstration of the Distributive Law for integer arithmetic (from cut-the-knot)

Chapter 6

Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binaryoperation on that set. It leaves other elements unchanged when combined with them. This is used for groups andrelated concepts.The term identity element is often shortened to identity (as will be done in this article) when there is no possibility ofconfusion.Let (S, ∗) be a set S with a binary operation ∗ on it. Then an element e of S is called a left identity if e ∗ a = a for alla in S, and a right identity if a ∗ e = a for all a in S. If e is both a left identity and a right identity, then it is called atwo-sided identity, or simply an identity.An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect tomultiplication is called a multiplicative identity (often denoted as 1). The distinction is used most often for sets thatsupport both binary operations, such as rings. The multiplicative identity is often called the unit in the latter context,where, though, a unit is often used in a broader sense, to mean an element with a multiplicative inverse.

6.1 Examples

6.2 Properties

As the last example (a semigroup) shows, it is possible for (S, ∗) to have several left identities. In fact, every elementcan be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a leftidentity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identityand r is a right identity then l = l ∗ r = r. In particular, there can never be more than one two-sided identity. If therewere two, e and f, then e ∗ f would have to be equal to both e and f.It is also quite possible for (S, ∗) to have no identity element. A common example of this is the cross product ofvectors; in this case, the absence of an identity element is related to the fact that the direction of any nonzero crossproduct is always orthogonal to any element multiplied – so that it is not possible to obtain a non-zero vector in thesame direction as the original. Another example would be the additive semigroup of positive natural numbers.

6.3 See also

• Absorbing element

• Additive inverse

• Inverse element

• Monoid

• Pseudo-ring

29

30 CHAPTER 6. IDENTITY ELEMENT

• Quasigroup

• Unital (disambiguation)

6.4 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p.14–15

Chapter 7

Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation toaddition, and a reciprocal in relation to multiplication. The intuition is of an element that can 'undo' the effect ofcombination with another given element. While the precise definition of an inverse element varies depending on thealgebraic structure involved, these definitions coincide in a group.The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'.

7.1 Formal definitions

7.1.1 In a unital magma

Let S be a set with a binary operation ∗ (i.e., a magma note that a magma also has closure under the binary operation).If e is an identity element of (S, ∗) (i.e., S is a unital magma) and a ∗ b = e , then a is called a left inverse of band b is called a right inverse of a . If an element x is both a left inverse and a right inverse of y , then x is calleda two-sided inverse, or simply an inverse, of y . An element with a two-sided inverse in S is called invertible inS . An element with an inverse element only on one side is left invertible, resp. right invertible. A unital magmain which all elements are invertible is called a loop. A loop whose binary operation satisfies the associative law is agroup.Just like (S, ∗) can have several left identities or several right identities, it is possible for an element to have severalleft inverses or several right inverses (but note that their definition above uses a two-sided identity e ). It can evenhave several left inverses and several right inverses.If the operation ∗ is associative then if an element has both a left inverse and a right inverse, they are equal. In otherwords, in a monoid (an associative unital magma) every element has at most one inverse (as defined in this section).In a monoid, the set of (left and right) invertible elements is a group, called the group of units of S , and denoted byU(S) or H1.A left-invertible element is left-cancellative, and analogously for right and two-sided.

7.1.2 In a semigroup

Main article: Regular semigroup

The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It’salso possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keepingassociativity, i.e. in a semigroup.In a semigroup S an element x is called (von Neumann) regular if there exists some element z in S such that xzx= x; z is sometimes called a pseudoinverse. An element y is called (simply) an inverse of x if xyx = x and y = yxy.Every regular element has at least one inverse: if x = xzx then it is easy to verify that y = zxz is an inverse of x asdefined in this section. Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, that

31

32 CHAPTER 7. INVERSE ELEMENT

is ee = e and ff = f. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and ex = xf = x, ye= fy = y, and e acts as a left identity on x, while f acts a right identity, and the left/right roles are reversed for y. Thissimple observation can be generalized using Green’s relations: every idempotent e in an arbitrary semigroup is a leftidentity for Re and right identity for Le.[1] An intuitive description of this fact is that every pair of mutually inverseelements produces a local left identity, and respectively, a local right identity.In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given inthis section. Only elements in the Green class H1 have an inverse from the unital magma perspective, whereas forany idempotent e, the elements of Hₑ have an inverse as defined in this section. Under this more general definition,inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then thesemigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly oneinverse as defined in this section, then the semigroup is called an inverse semigroup. Finally, an inverse semigroupwith only one idempotent is a group. An inverse semigroup may have an absorbing element 0 because 000 = 0,whereas a group may not.Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse. This isgenerally justified because in most applications (e.g. all examples in this article) associativity holds, which makes thisnotion a generalization of the left/right inverse relative to an identity.

7.1.3 U-semigroups

A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (a°)° = afor all a in S; this endows S with a type ⟨2,1⟩ algebra. A semigroup endowed with such an operation is called aU-semigroup. Although it may seem that a° will be the inverse of a, this is not necessarily the case. In order toobtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classesof U-semigroups have been studied:[2]

• I-semigroups, in which the interaction axiom is aa°a = a

• *-semigroups, in which the interaction axiom is (ab)° = b°a°. Such an operation is called an involution, andtypically denoted by a*

Clearly a group is both an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theoryare completely regular semigroups; these are I-semigroups in which one additionally has aa° = a°a; in other wordsevery element has commuting pseudoinverse a°. There are few concrete examples of such semigroups however; mostare completely simple semigroups. In contrast, a subclass of *-semigroups, the *-regular semigroups (in the senseof Drazin), yield one of best known examples of a (unique) pseudoinverse, the Moore–Penrose inverse. In this casehowever the involution a* is not the pseudoinverse. Rather, the pseudoinverse of x is the unique element y suchthat xyx = x, yxy = y, (xy)* = xy, (yx)* = yx. Since *-regular semigroups generalize inverse semigroups, the uniqueelement defined this way in a *-regular semigroup is called the generalized inverse or Penrose–Moore inverse.

7.1.4 Rings and semirings

Main article: Quasiregular element

7.2 Examples

All examples in this section involve associative operators, thus we shall use the terms left/right inverse for the unitalmagma-based definition, and quasi-inverse for its more general version.

7.2.1 Real numbers

Every real number x has an additive inverse (i.e. an inverse with respect to addition) given by −x . Every nonzeroreal number x has a multiplicative inverse (i.e. an inverse with respect to multiplication) given by 1

x (or x−1 ). Bycontrast, zero has no multiplicative inverse, but it has a unique quasi-inverse, “0(ZERO)" itself.

7.2. EXAMPLES 33

7.2.2 Functions and partial functions

A function g is the left (resp. right) inverse of a function f (for function composition), if and only if g ◦f (resp. f ◦g) is the identity function on the domain (resp. codomain) of f . The inverse of a function f is often written f−1 , butthis notation is sometimes ambiguous. Only bijections have two-sided inverses, but any function has a quasi-inverse,i.e. the full transformation monoid is regular. The monoid of partial functions is also regular, whereas the monoid ofinjective partial transformations is the prototypical inverse semigroup.

7.2.3 Galois connections

The lower and upper adjoints in a (monotone) Galois connection, L and G are quasi-inverses of each other, i.e. LGL= L and GLG = G and one uniquely determines the other. They are not left or right inverses of each other however.

7.2.4 Matrices

A square matrix M with entries in a field K is invertible (in the set of all square matrices of the same size, undermatrix multiplication) if and only if its determinant is different from zero. If the determinant of M is zero, it isimpossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the otherone. See invertible matrix for more.More generally, a square matrix over a commutative ring R is invertible if and only if its determinant is invertible inR .Non-square matrices of full rank have several one-sided inverses:[3]

• For A : m× n | m > n we have a left inverse: (ATA)−1AT︸ ︷︷ ︸A−1

left

A = In

• For A : m× n | m < n we have a right inverse: AAT (AAT )−1︸ ︷︷ ︸A−1

right

= Im

The left inverse can be used to determine the least norm solution of Ax = b , which is also the least squares formulafor regression and is given by x = (ATA)−1AT b.

No rank deficient matrix has any (even one-sided) inverse. However, the Moore–Penrose pseudoinverse exists for allmatrices, and coincides with the left or right (or true) inverse when it exists.As an example of matrix inverses, consider:

A : 2× 3 =

[1 2 34 5 6

]So, as m < n, we have a right inverse, A−1

right = AT (AAT )−1. By components it is computed as

AAT =

[1 2 34 5 6

]·

1 42 53 6

=

[14 3232 77

]

(AAT )−1 =

[14 3232 77

]−1

=1

54

[77 −32−32 14

]

AT (AAT )−1 =1

54

1 42 53 6

·[77 −32−32 14

]=

1

18

−17 8−2 213 −4

= A−1right

The left inverse doesn't exist, because

34 CHAPTER 7. INVERSE ELEMENT

ATA =

1 42 53 6

·[1 2 34 5 6

]=

17 22 2722 29 3627 36 45

which is a singular matrix, and cannot be inverted.

7.3 See also• Loop (algebra)

• Division ring

• Unit (ring theory)

• Latin square property

7.4 Notes[1] Howie, prop. 2.3.3, p. 51

[2] Howie p. 102

[3] MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse.

7.5 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p.15 (def in unital magma) and p. 33 (def in semigroup)

• Howie, JohnM. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN0-19-851194-9. containsall of the semigroup material herein except *-regular semigroups.

• Drazin, M.P., Regular semigroups with involution, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46

• Miyuki Yamada, P-systems in regular semigroups, Semigroup Forum, 24(1), December 1982, pp. 173–187

• Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377.

Chapter 8

Power associativity

In abstract algebra, power associativity is a property of a binary operation which is a weak form of associativity.An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element isassociative. Concretely, this means that if an element x is multiplied by itself several times, it doesn't matter in whichorder the multiplications are carried out, so for instance x(x(xx)) = (x(xx))x = (xx)(xx). This is stronger than poweralternativity, that is (xx)x = x(xx) for every x in the algebra, but weaker than alternativity or associativity.Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which arenon-associative) and even some non-alternative algebras like the sedenions and Okubo algebras. Any algebra whoseelements are idempotent is also power-associative.Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whether x3 should be defined as (xx)x or as x(xx), since theseare equal. Exponentiation to the power of zero can also be defined if the operation has an identity element, so theexistence of identity elements is useful in power-associative contexts.A substitution law holds for real power-associative algebras with unit, which basically asserts that multiplication ofpolynomials works as expected. For f a real polynomial in x, and for any a in such an algebra define f(a) to be theelement of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f andg, we have that (fg)(a) = f(a)g(a).

8.1 References• Albert, A. Adrian (1948). “Power-associative rings”. Transactions of the American Mathematical Society 64:552–593. doi:10.2307/1990399. ISSN 0002-9947. JSTOR 1990399. MR 0027750. Zbl 0033.15402.

• Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The book of involutions.Colloquium Publications 44. With a preface by J. Tits. Providence, RI: AmericanMathematical Society. ISBN0-8218-0904-0. Zbl 0955.16001.

• Okubo, Susumu (1995). Introduction to octonion and other non-associative algebras in physics. MontrollMemorial Lecture Series in Mathematical Physics 2. Cambridge University Press. p. 17. ISBN 0-521-01792-0. MR 1356224. Zbl 0841.17001.

• Schafer, R.D. (1995) [1966]. An introduction to non-associative algebras. Dover. pp. 128–148. ISBN 0-486-68813-5.

35

Chapter 9

Quasi-commutative property

In mathematics, the quasi-commutative property is an extension or generalization of the general commutativeproperty. This property is used in certain specific applications with various definitions.

9.1 Applied to matrices

Two matrices p and q are said to have the commutative property whenever

pq = qp

The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices x and y

xy − yx = z

satisfy the quasi-commutative property whenever z satisfies the following properties:

xz = zx

yz = zy

An example is found in the matrix mechanics introduced by Heisenberg as a version of quantum mechanics. In thismechanics, p and q are infinite matrices corresponding respectively to the momentum and position variables of aparticle.[1] These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite unitmatrix, where ħ is the reduced Planck constant.

9.2 Applied to functions

A function f, defined as follows:

f : X × Y → X

is said to be quasi-commutative[2] if for all x ∈ X and for all y1, y2 ∈ Y ,

f(f(x, y1), y2) = f(f(x, y2), y1)

36

9.3. SEE ALSO 37

9.3 See also• Commutative property

• Accumulator (cryptography)

9.4 References[1] Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.

[2] Benaloh, J., & De Mare, M. (1994, January). One-way accumulators: A decentralized alternative to digital signatures. InAdvances in Cryptology—EUROCRYPT’93 (pp. 274–285). Springer Berlin Heidelberg.

38 CHAPTER 9. QUASI-COMMUTATIVE PROPERTY

9.5 Text and image sources, contributors, and licenses

9.5.1 Text• Absorbing element Source: https://en.wikipedia.org/wiki/Absorbing_element?oldid=695749427 Contributors: Giftlite, Creidieki, Zaak,Salix alba, Melchoir, Krishnachandranvn, Don4of4, He7d3r, ThatWikiGuy, Addbot, Yobot, Pcap, Loveless, FrescoBot, Horcrux92,EmausBot, ZéroBot, ClueBot NG, Helpful Pixie Bot, J58660, BemusedObserver and Anonymous: 6

• Anticommutativity Source: https://en.wikipedia.org/wiki/Anticommutativity?oldid=654287577 Contributors: Patrick, Karada, Docu,Silverfish, Phys, BenRG, Robbot, Fotoplasma, Paul August, Rgdboer, Susvolans, Sam Korn, InShaneee, Mindmatrix, Isnow, Salixalba, HappyCamper, RussBot, Melchoir, Leland McInnes, Ohconfucius, 16@r, Loadmaster, Dan Gluck, Salgueiro~enwiki, Error792,Daniele.tampieri, DusanJ, TXiKiBoT,AnonymousDissident, SieBot, JP.Martin-Flatin, Addbot, HerculeBot, Luckas-bot, Miracleworker5263,FrescoBot, EmausBot, Wham Bam Rock II, Quondum, Irisrune, Guy vandegrift, Justincheng12345-bot and Anonymous: 12

• Associative property Source: https://en.wikipedia.org/wiki/Associative_property?oldid=704510167 Contributors: AxelBoldt, Zundark,Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Patrick, Michael Hardy, Ellywa, Andres, Pizza Puzzle,Ideyal, Charles Matthews, Dysprosia, Kwantus, Robbot, Mattblack82, Wikibot, Wereon, Robinh, Tobias Bergemann, Giftlite, Smjg,Lethe, Herbee, Brona, Jason Quinn, Deleting Unnecessary Words, Creidieki, Rich Farmbrough, Guanabot, Paul August, Rgdboer, Jum-buck, Alansohn, Gary, Burn, H2g2bob, Bsadowski1, Oleg Alexandrov, Linas, Palica, Graham87, Deadcorpse, SixWingedSeraph, Fre-plySpang, Yurik, Josh Parris, Rjwilmsi, Salix alba, Mathbot, Chobot, YurikBot, Thane, AdiJapan, BOT-Superzerocool, Bota47, Banus,KnightRider~enwiki, Mmernex, Melchoir, PJTraill, Thumperward, Fuzzform, SchfiftyThree, Octahedron80, Zven, Furby100, Wine Guy,Smooth O, Cybercobra, Cothrun, Will Beback, SashatoBot, Frentos, IronGargoyle, Physis, 16@r, JRSpriggs, CBM, Gregbard, Xtv, MrGronk, Rbanzai, Thijs!bot, Egriffin, Marek69, Escarbot, Salgueiro~enwiki, DAGwyn, David Eppstein, JCraw, Anaxial, Trusilver, Acala-mari, Pyrospirit, Krishnachandranvn, Daniel5Ko, GaborLajos, Darklich14, LokiClock, AlnoktaBOT, Rei-bot, Wiae, SieBot, Gerakibot,Flyer22 Reborn, Hello71, Trefgy, Classicalecon, ClueBot, The Thing That Should Not Be, Cliff, Razimantv, Mudshark36, Auntof6,DragonBot, Watchduck, Bender2k14, SoxBot III, Addbot, Some jerk on the Internet, Download, BepBot, Tide rolls, Jarble, Luckas-bot,Yobot, TaBOT-zerem, Kuraga, MauritsBot, GrouchoBot, PI314r, Charvest, Ex13, Erik9bot, MacMed, Pinethicket, MastiBot, Fox Wil-son, Ujoimro, GoingBatty, ZéroBot, NuclearDuckie, Quondum, D.Lazard, Donner60, EdoBot, Crown Prince, ClueBot NG, Wcherowi,Matthiaspaul, Kevin Gorman, Widr, BG19bot, Furkaocean, IkamusumeFan, SuperNerd137, Rang213, Stephan Kulla, Lugia2453, Jsh-flynn, Epicgenius, Kenrambergm2374, Matthew Kastor, Subcientifico, Monarchrob1, Allthefoxes and Anonymous: 126

• Commutative property Source: https://en.wikipedia.org/wiki/Commutative_property?oldid=704664007 Contributors: AxelBoldt, Zun-dark, Tarquin, Andre Engels, Christian List, Toby~enwiki, Toby Bartels, Patrick, Michael Hardy, Wshun, Ixfd64, GTBacchus, Ahoerste-meier, Snoyes, Jll, Pizza Puzzle, Ideyal, Charles Matthews, Wikiborg, Dysprosia, Jeffq, Robbot, RedWolf, Romanm, Robinh, Isopropyl,Tobias Bergemann, Enochlau, Giftlite, BenFrantzDale, Lupin, Herbee, Peruvianllama, Waltpohl, Frencheigh, Gdr, Knutux, OverlordQ,B.d.mills, Chris Howard, Mormegil, Rich Farmbrough, Ebelular, Mikael Brockman, Dbachmann, Paul August, MisterSheik, El C, Szquir-rel, Touriste, Samadam, Malcolm rowe, Jumbuck, Arthena, Mattpickman, Mlessard, Burn, Mlm42, Stillnotelf, Tony Sidaway, OlegAlexandrov, The JPS, Linas, Justinlebar, Jeff3000, Palica, Ashmoo, Graham87, Josh Parris, Rjwilmsi, Salix alba, Vegaswikian, FlaBot,VKokielov, Ground Zero, Srleffler, Masnevets, Reetep, YurikBot, Hairy Dude, Wolfmankurd, Michael Slone, Rick Norwood, SamuelHuang, Derek.cashman, FF2010, Petri Krohn, Vicarious, SmackBot, YellowMonkey, Slashme, Melchoir, KocjoBot~enwiki, Jab843,PJTraill, Chris the speller, Bluebot, Master of Puppets, Thumperward, SchfiftyThree, Complexica, Octahedron80, DHN-bot~enwiki,JustUser, Cybercobra, Wybot, Thehakimboy, Acdx, Bando26, 16@r, A. Parrot, Childzy, Dan Gluck, Iridescent, Dreftymac, DBooth,Robert.McGibbon, Floridi~enwiki, Unmitigated Success, Gregbard, MichaelRWolf, Cydebot, Larsnostdal, Kozuch, Dinominant, Jame-sAM,Headbomb, SecondQuantization,Wmasterj, Thomprod, Dzer0, Grayshi, Escarbot, Fr33ke, AntiVandalBot, Nacho Librarian, Gcm,100110100, Mikemill, Wikidudeman, DAGwyn, Dirac66, JoergenB, MartinBot, Nev1, Daniele.tampieri, Haseldon, Policron, KylieTas-tic, Sarregouset, Useight, VolkovBot, TreasuryTag, Am Fiosaigear~enwiki, Philip Trueman, TXiKiBoT, Oshwah, Anonymous Dissident,Aaron Rotenberg, Geometry guy, Spinningspark, Life, Liberty, Property, SieBot, Ivan Štambuk, Legion fi, Toddst1, Flyer22 Reborn,Xvani, Weston.pace, OKBot, Mike2vil, Francvs, Classicalecon, ClueBot, Cliff, Bloodholds, R000t, CounterVandalismBot, Deathnomad,Excirial, Ftbhrygvn, Joe8824, Nafis ru, Stephen Poppitt, Addbot, LaaknorBot, SpBot, Gail, Luckas-bot, Yobot, Weisicong, AnomieBOT,DemocraticLuntz, Citation bot, MauritsBot, Xqbot, Dithridge, 12cookk, Ubcule, GrouchoBot, Omnipaedista, Dger, HamburgerRadio,MacMed, Pinethicket, Adlerbot, Serols, Psimmler, ThinkEnemies, JV Smithy, Onel5969, Ujoimro, Aceshooter, Slightsmile, Checkingfax,Quondum, Joshlepaknpsa, Wayne Slam, Arnaugir, Scientific29, ClueBot NG, Wcherowi, Gilderien, Sayginer, Marechal Ney, Widr, Avo-catoBot, MarkArsten, Ameulen11, CeraBot, ChrisGualtieri, None but shining hours, Khazar2, Dexbot, StephanKulla, Fox2k11, Stewwie,Dskjhgds, DavidLeighEllis, Spacenut42, Davidliu0421, Wikibritannica, Niallhoranluv123, JMP EAX, Troolium, Hinmatóowyalahtqit,Holt Mcdougal, 3 of Diamonds, Fazbear7891 and Anonymous: 188

• Distributive property Source: https://en.wikipedia.org/wiki/Distributive_property?oldid=702722619 Contributors: AxelBoldt, Tarquin,Youssefsan, TobyBartels, Patrick, Xavic69,Michael Hardy, Andres, Ideyal, Dysprosia, Malcohol, Andrewman327, Shizhao, PuzzletChung,Romanm, Chris Roy,Wikibot, Tobias Bergemann, Giftlite, Markus Krötzsch, Dissident, Nodmonkey, Mike Rosoft, Smimram, Discospin-ster, Paul August, ESkog, Rgdboer, EmilJ, Bobo192, Robotje, Smalljim, Jumbuck, Arthena, Keenan Pepper, Mykej, Bsadowski1, Blax-thos, Linas, Evershade, Isnow, Marudubshinki, Salix alba, Vegaswikian, Nneonneo, Bfigura, FlaBot, Alexb@cut-the-knot.com, Mathbot,Andy85719, Ichudov, DVdm, YurikBot, Michael Slone, Grafen, Trovatore, Bota47, Banus, Melchoir, Yamaguchi , Bluebot, Ladislavthe Posthumous, Octahedron80, UNV, Jiddisch~enwiki, Khazar, FrozenMan, Bando26, 16@r, Dicklyon, EdC~enwiki, Engelec, Exza-kin, Jokes Free4Me, Simeon, Gregbard, Thijs!bot, Barticus88, Marek69, Nezzadar, Escarbot, Mhaitham.shammaa, Salgueiro~enwiki,JAnDbot, Onkel Tuca~enwiki, Acroterion, Numbo3, Katalaveno, AntiSpamBot, GaborLajos, Lyctc, Idioma-bot, Janice Margaret Vian,Montchav, TXiKiBoT, Oshwah, Anonymous Dissident, Dictouray, Oxfordwang, Martin451, Skylarkmichelle, Jackfork, Enviroboy,Dmcq, AlleborgoBot, Gerakibot, Bentogoa, Flyer22 Reborn, Radon210, Hello71, ClueBot, The Thing That Should Not Be, Cliff, MildBill Hiccup, Niceguyedc, Goldkingtut5, Excirial, Jusdafax, NuclearWarfare, NERIC-Security, Pichpich, Mm40, Addbot, Jojhutton, Ron-hjones, Zarcadia, Favonian, Squandermania, Jarble, Ben Ben, Legobot, Luckas-bot, AnomieBOT, Materialscientist, NFD9001, Great-fermat, False vacuum, RibotBOT, Pinethicket, I dream of horses, MastiBot, Andrea105, Slon02, Saul34, J36miles, John of Reading,Davejohnsan, Orphan Wiki, Super48paul, Sp33dyphil, Slawekb, Quondum, BrokenAnchorBot, TyA, Donner60, Chewings72, DASH-BotAV, AlecJansen, ClueBot NG, Wcherowi, IfYouDoIfYouDon't, Dreth, O.Koslowski, Asukite, Widr, Vibhijain, Helpful Pixie Bot,Pmi1924, BG19bot, TCN7JM, Dan653, Forkloop, CallofDutyboy9, EuroCarGT, Sandeep.ps4, Christian314, Ivashikhmin, Israphel-Mac, Mogism, Makecat-bot, Lugia2453, Gphilip, Brirush, Wywin, BB-GUN101, ElHef, DavidLeighEllis, Shaun9876, Pkramer2021,Kitkat1234567880, Kcolemantwin3, Gracecandy1143, David88063, Abruce123412, Amortias, Solid Frog, Jj 1213 wiki, Dalangster,Iwamwickham, 123456me123456, Some1Redirects4You, ProprioMe OW, Friendlyyoshi, Pockybits and Anonymous: 254

9.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 39

• Identity element Source: https://en.wikipedia.org/wiki/Identity_element?oldid=692628344 Contributors: AxelBoldt, Zundark, XJaM,Toby~enwiki, Toby Bartels, Tim Starling, SebastianHelm, Pcb21, Ellywa, Andres, Ideyal, Charles Matthews, Dysprosia, Hyacinth, DavidShay, Sandman~enwiki, Robbot, Mattblack82, Tobias Bergemann, Giftlite, Lethe, Leonard G., Gubbubu, Paul August, EmilJ, Dreish,Obradovic Goran, Helix84, Varuna, Jumbuck, Jérôme, Eric Kvaalen, ABCD, Woodstone, Mindmatrix, Bkkbrad, MattGiuca, Salix alba,RobertG, Mathbot, Chobot, YurikBot, Philip Hazelden, GLaDOS, Bota47, Haza-w, Incnis Mrsi, Bggoldie~enwiki, Melchoir, Uny-oyega, BiT, Octahedron80, Jiddisch~enwiki, Bombshell, Jim.belk, TooMuchMath, Luokehao, EdC~enwiki, Aeons, Vaughan Pratt, JokesFree4Me, Neelix, Cydebot, Riojajar~enwiki, Thijs!bot, Xyctka, Acroterion, MetsBot, R'n'B, Beremiz~enwiki, Philip Trueman, Davehi1,Wiae, SieBot, Gerakibot, This, that and the other, Keilana, Belinrahs, Ctxppc, OKBot, ClueBot, SilvonenBot, Addbot, Jarble, Luckas-bot,2D, TaBOT-zerem, Pcap, AnomieBOT, Xqbot, WaysToEscape, LucienBOT, Baumgaertner, EmausBot, Chricho, Jay-Sebastos, EdoBot,ClueBot NG, Helpful Pixie Bot, Dan653, Mikelam98, Sajjad.r, Mr. Guye, Razibot, GigaGerard, Yamaha5, ANGEL-T13, Rishabhlodu,MfR, Sweepy and Anonymous: 49

• Inverse element Source: https://en.wikipedia.org/wiki/Inverse_element?oldid=704104529 Contributors: Zundark, Michael Hardy, Iso-morphic, Charles Matthews, Timwi, Dysprosia, David Shay, MathMartin, Tobias Bergemann, Tosha, Giftlite, BenFrantzDale, Fropuff,Phe, Jacob grace, Paul August, Cmdrjameson, Obradovic Goran, Helix84, HasharBot~enwiki, ABCD, MFH, OneWeirdDude, Salixalba, FlaBot, Tardis, Chobot, YurikBot, ManoaChild, Bota47, Reyk, SmackBot, Incnis Mrsi, Melchoir, LVC, Moocowpong1, Gelingvis-toj, Oli Filth, Rludlow, Saippuakauppias, Mets501, Rschwieb, Myasuda, Kilva, Beta16, Laser.Y, Magioladitis, David Eppstein, Toluno,DavidCBryant, Idioma-bot, TXiKiBoT, Beusson, Neparis, JackSchmidt, Marino-slo, Digitizdat, Rhuso, Addbot, Abovechief, Luckas-bot, Ht686rg90, Pcap, Ciphers, Xqbot, EnderWiggin1, WikitanvirBot, ZéroBot, ClueBot NG, Helpful Pixie Bot, ElphiBot, AvocatoBot,Deltahedron, JMP EAX, Loraof, Some1Redirects4You and Anonymous: 24

• Power associativity Source: https://en.wikipedia.org/wiki/Power_associativity?oldid=705105980 Contributors: AxelBoldt, Bryan Derk-sen, Zundark, Toby Bartels, Michael Hardy, Loren Rosen, Charles Matthews, VivaEmilyDavies, Isnow, Graham87, Jshadias, Rjwilmsi,R.e.b., Michael Slone, Melchoir, Rschwieb, Cydebot, Headbomb, WinBot, .anacondabot, Vanish2, SieBot, Ersik, Legobot, Dewey pro-cess, Ethaniel, Quondum, Wabbott9, Vladimirdx, Deltahedron and Anonymous: 8

• Quasi-commutative property Source: https://en.wikipedia.org/wiki/Quasi-commutative_property?oldid=695775895Contributors: MichaelHardy, Dirac66, Marius siuram, OccultZone and Hydronium Hydroxide

9.5.2 Images• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)

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