# Inverse element

In mathematics , inverse elements appear when studying algebraic structures . Such a structure consists of a set and a two-digit link (arithmetic operation) defined in it . In this context this means: If you combine any element of the set and its inverse with the arithmetic operation, you always get the so-called neutral element as the result.

Colloquially, the inverse element could also be called the “reverse” or “opposite” element. However, one must not forget in which context one is, because there are a multitude of possibilities to define a set or an arithmetic operation (which one usually does not know from school mathematics).

## definition

Be a set with a two-digit link and a neutral element . Be . ${\ displaystyle A}$ ${\ displaystyle \ circ}$ ${\ displaystyle e}$${\ displaystyle a, b \ in A}$

If there is initially no commutativity, i. H. it is only true that it is called right-invertible with the right-inverse element , and it is called left-invertible with the left-inverse element . ${\ displaystyle a \ circ b = e}$${\ displaystyle a}$ ${\ displaystyle b}$${\ displaystyle b}$ ${\ displaystyle a}$

If, on the other hand, an element with exists for an element , then only means invertible or invertible on both sides with the inverse element . ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a \ circ b = b \ circ a = e}$${\ displaystyle a}$ ${\ displaystyle b}$

A double-sided inverse element is often written as with additive notation of the link , with multiplicative notation often as . ${\ displaystyle (-a)}$${\ displaystyle a ^ {- 1}}$

## properties

The link is assumed to be associative , i.e. H. be a monoid . ${\ displaystyle \ circ}$${\ displaystyle A}$

• If an element is both left and right invertible, then all left and right inverse elements of match. In particular, it is invertible on both sides, and the element which is inverse to an element which is invertible on both sides is clearly determined.${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle a}$
• The inverse of the inverse is the original element, so . The single-digit link is therefore an involution on the set of elements that can be inverted on both sides.${\ displaystyle \ left (a ^ {- 1} \ right) ^ {- 1} = a}$ ${\ displaystyle ^ {- 1}}$
• If a product can be right-inverted, it can also be right-inverted; is left-invertible, so is also left-inverted. If and are invertible on both sides, so too , and it applies${\ displaystyle a \ circ b}$${\ displaystyle a}$${\ displaystyle a \ circ b}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a \ circ b}$
${\ displaystyle (a \ circ b) ^ {- 1} = b ^ {- 1} \ circ a ^ {- 1}.}$
This property is sometimes sock-shoe rule (English: shoe (s) -socks property ) or Shirt Jacket usually called: When taking off shoes and socks or shirt and jacket have to reverse the order of tightening.
• The mutually invertible elements of a monoid form a group. This follows from the two previous properties. This group is called the unit group. This term is especially common when speaking of the multiplicative monoid of a unitary ring .
• A monoid homomorphism maps inverse to inverse; i.e., is invertible, then is also invertible, and it holds${\ displaystyle f \ colon A \ to B}$${\ displaystyle a \ in A}$${\ displaystyle f \ left (a \ right) \ in B}$
${\ displaystyle f \ left (a ^ {- 1} \ right) = f \ left (a \ right) ^ {- 1}.}$

If the associative law does not apply in general to an algebraic structure with a neutral element, it is possible that an element has several left inverses and several right inverses. ${\ displaystyle \ left (A, \ circ \ right)}$

## Examples

In the known sets of numbers ( natural numbers including the zero 0, rational numbers , etc.) one has an addition with the neutral element 0. The additive inverse of a number is the number which, when added, results in 0, i.e. its opposite or also its opposite number . If you add to a term , you add a so-called constructive or productive zero . ${\ displaystyle a}$${\ displaystyle a}$ ${\ displaystyle -a}$
${\ displaystyle a + (- a) = aa = 0}$

For example, is the opposite of , for . For the same reason, the opposite of turn is , so is . This generally applies to all numbers. ${\ displaystyle -7}$${\ displaystyle 7}$${\ displaystyle 7 + (- 7) = (- 7) + 7 = 0}$${\ displaystyle -7}$${\ displaystyle 7}$${\ displaystyle - (- 7) = 7}$

Therefore, the opposite of a number is not always a negative number , i.e. a number . The following applies to negative numbers : d. H. the opposite of a negative number is a positive number . However, the opposite of a positive number is always a negative number. ${\ displaystyle a <0}$${\ displaystyle a}$${\ displaystyle -a> 0,}$

The opposite is always obtained in these cases by multiplying by −1, i.e. H. . ${\ displaystyle -a = -1 \ cdot a}$

In general, the additively inverse element regularly exists in additively written Abelian groups ${\ displaystyle (G, +)}$ . The main examples of this are:

In addition, there are number sets in which an addition can be carried out, but in which there are no additively inverse elements . Such are z. B.

The whole numbers can be constructed from the natural numbers by formally adding the negatives (and 0, if 0 is not defined as a natural number ) and defining suitable calculation rules. Seen in this way, every natural number has an opposite that is at the same time its negative. However, since this is not a natural number (except for 0, if 0 is defined as a natural number), the set of natural numbers is not closed under the opposition or subtraction (addition with an opposite).

### Multiplicative inverse

In the above-mentioned number sets there is also a multiplication with neutral element 1. The multiplicative inverse of a number a is the number associated with a multiplied results. 1 So it is the reciprocal of a .

For example, the reciprocal of 7 is the rational number 1/7; however, in integers, 7 has no multiplicative inverse.

If a ring R is given in general , then the elements that have multiplicative inverses are called units of the ring. In the theory of divisibility , no distinction is usually made between ring elements that differ by one unit multiplicatively (i.e. a = eb with one unit e ).

In remainder class rings , the multiplicative inverse can be calculated using the extended Euclidean algorithm , if it exists.

### Inverse function

Consider the set of all functions from a lot of . On this set one has the composition (execution one after the other) as a link, defined by ${\ displaystyle A ^ {A}}$${\ displaystyle f \ colon A \ to A}$${\ displaystyle A}$${\ displaystyle A}$

${\ displaystyle g \ circ f \ colon \, A \ to A, \, a \ mapsto (g \ circ f) (a): = g (f (a))}$.

The composition is associative and has the identical image as a neutral element. ${\ displaystyle \ operatorname {id} _ {A} \ colon A \ to A, \, a \ mapsto a}$

If a function is bijective , then the inverse function is the inverse element of in . ${\ displaystyle f \ colon A \ to A}$ ${\ displaystyle f ^ {- 1} \ colon A \ to A}$${\ displaystyle f}$${\ displaystyle A ^ {A}}$

One generalizes this term to bijective functions and gets an inverse function with and${\ displaystyle f \ colon A \ to B}$${\ displaystyle f ^ {- 1} \ colon B \ to A}$${\ displaystyle f ^ {- 1} \ circ f = \ operatorname {id} _ {A}}$${\ displaystyle f \ circ f ^ {- 1} = \ operatorname {id} _ {B}.}$

Is A a body such as B. the real numbers , then one must not confuse the inverse function with the reciprocal value ! The inverse function is only defined when is bijective, and the reciprocal is only defined when has no zeros . Even if a subset of bijective maps to itself, the inverse function and reciprocal value generally do not match. ${\ displaystyle f ^ {- 1}}$${\ displaystyle 1 / f}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle \ mathbb {R} \ setminus \ {0 \}}$

For example, the function has an inverse function and an inverse , but they do not match. (Where is the set of positive real numbers.) ${\ displaystyle f \ colon \ mathbb {R} ^ {+} \ to \ mathbb {R} ^ {+}, \, x \ mapsto x ^ {2}}$${\ displaystyle f ^ {- 1} \ colon \ mathbb {R} ^ {+} \ to \ mathbb {R} ^ {+}, \, x \ mapsto {\ sqrt {x}}}$${\ displaystyle \ left ({\ frac {1} {f}} \ right) (x) = {\ frac {1} {f (x)}} = {\ frac {1} {x ^ {2}} }}$${\ displaystyle \ mathbb {R} ^ {+} = (0, \ infty)}$

## Self-inverse elements

In a monoid with the neutral element , an element is called self-inverse if: ${\ displaystyle (M, *)}$${\ displaystyle e \ in M}$${\ displaystyle a \ in M}$

${\ displaystyle a * a = e \ {\ text {or}} \ a ^ {{-} 1} = a}$
• The neutral element is self-inverse in every monoid: ${\ displaystyle e * e = e}$
• In a link table for a monoid, the self-inverse elements can be recognized by the fact that the neutral element is on the diagonal.
• Example:
${\ displaystyle *}$ e a b c
e e a b c
a a e c b
b b c e a
c c b a e
• A monoid in which every element is self-inverse is always a commutative group.
• Proof:
Since each element has an inverse element (namely itself), the monoid is a group.
As with too , is self-inverse, such that ${\ displaystyle a, b \ in M}$${\ displaystyle (a * b) \ in M}$${\ displaystyle (a * b)}$
${\ displaystyle (a * b) * (a * b) = e}$
But also applies (because of the associative law)
${\ displaystyle (a * b) * (b * a) = a * (b * (b * a)) = a * ((b * b) * a) = a * (e * a) = a * a = e.}$
Because of the uniqueness of the (right) inverse element in a group (see above) must therefore apply
${\ displaystyle b * a = a * b.}$

## Generalization: Definitions without a neutral element

Inverse elements can also be defined without the existence of a neutral element, i.e. in any magma or a semigroup .

### (weak) inverses in a magma

Is there a unique element in any magma for a , so that applies to all : ${\ displaystyle (M, *)}$${\ displaystyle a \ in M}$${\ displaystyle a ^ {{-} 1} \ in M}$${\ displaystyle b \ in M}$

${\ displaystyle a ^ {{-} 1} * (a * b) = b = (b * a) * a ^ {{-} 1},}$

then one calls (weak) invertible and the (weak) inverse of . A magma in which all can be inverted (weak) that has inverse property (Engl. Inverse property ), and is called then quasigroup inverse property . ${\ displaystyle a}$ ${\ displaystyle a ^ {{-} 1}}$${\ displaystyle a}$${\ displaystyle (M, *),}$${\ displaystyle a \ in M}$${\ displaystyle (M, *),}$

A quasi-group with inverse property is a quasi-group (proof see quasi-group ). A semi-group that has the inverse property is therefore already a group .

According to this definition, and operate together like a neutral element on each element , but there does not have to be an explicit, neutral element. ${\ displaystyle a}$${\ displaystyle a ^ {{-} 1}}$${\ displaystyle b \ in M}$

In a semi-group that has the inverse property , however, due to the associative law, applies to all : ${\ displaystyle (M, *),}$${\ displaystyle b \ in M}$

${\ displaystyle (a ^ {{-} 1} * a) * b = a ^ {{-} 1} * (a * b) = b = (b * a) * a ^ {{-} 1} = b * (a * a ^ {{-} 1}),}$

So the (unambiguous) neutral element of In (semi-) groups, so both definitions of inverse elements agree, in quasi-groups not necessarily. ${\ displaystyle e: = (a ^ {{-} 1} * a)}$${\ displaystyle (M, *).}$

### (crossed over) Inverse in a magma

Are there any Magma for an element , such that for all true: ${\ displaystyle (M, *)}$${\ displaystyle a \ in M}$${\ displaystyle a ^ {{-} 1} \ in M}$${\ displaystyle b \ in M}$

${\ displaystyle a ^ {{-} 1} * (b * a) = b = (a * b) * a ^ {{-} 1},}$

it is called (cross) can be inverted and a (cross) inverse (Engl. crossed inverse ) of . ${\ displaystyle a}$ ${\ displaystyle a ^ {{-} 1}}$${\ displaystyle a}$

A magma where all a (cross) Inverse who served About Cross-inverse property (Engl. Crossed inverse property , CIP), and is called then CIP Magma (Engl. CIP groupoid ). ${\ displaystyle (M, *),}$${\ displaystyle a \ in M}$${\ displaystyle (M, *),}$

In a CIP magma the (crossed over) inverse for an element is uniquely determined. In addition, a CIP magma is always a quasi-group ( CIP quasi-group ).

An Abelian group has the crossed inverse property, a non-commutative group does not necessarily:

${\ displaystyle (b * a) = (a * b) \ implies a ^ {{-} 1} * (b * a) = a ^ {{-} 1} * (a * b) = (a ^ { {-} 1} * a) * b = e * b = b \ land (a * b) * a ^ {{-} 1} = (b * a) * a ^ {{-} 1} = b * (a * a ^ {{-} 1}) = b * e = b}$

### (relative) inverse in a semigroup

In an inverse semigroup a is (relative) Inverse (engl. Relative inverse ) to a defined by the fact that: ${\ displaystyle (A, *)}$${\ displaystyle y \ in A}$${\ displaystyle x \ in A}$

${\ displaystyle x * y * x = x}$and .${\ displaystyle y * x * y = y}$

This definition is even weaker than in a quasi-group with inverse property, since otherwise the inverse semigroup would already be a group.