Inverse element

A partial operation is associative if

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

for every x, y, z in X for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined.

Examples of non-total associative operations are multiplication of matrices of arbitrary size, and function composition.

Let ${\displaystyle *}$ be a possibly partial associative operation on a set X.

An identity element, or simply an identity is an element e such that

${\displaystyle x*e=x\quad {\text{and}}\quad e*y=y}$

for every x and y for which the left-hand sides of the equalities are defined.

If e and f are two identity elements such that ${\displaystyle e*f}$ is defined, then ${\displaystyle e=f.}$ (This results immediately from the definition, by ${\displaystyle e=e*f=f.}$)

It follows that a total operation has at most one identity element, and if e and f are different identities, then ${\displaystyle e*f}$ is not defined.

For example, in the case of matrix multiplication, there is one n×n identity matrix for every positive integer n, and two identity matrices of different size cannot be multiplied together.

Similarly, identity functions are identity elements for function composition, and the composition of the identity functions of two different sets are not defined.

If ${\displaystyle x*y=e,}$ where e is an identity element, one says that x is a left inverse of y, and y is a right inverse of x.

Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on nonnegative integers, which has 0 as additive identity, and 0 is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.

An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the functions from the integers to the integers. The doubling function ${\displaystyle x\mapsto 2x}$ has infinitely many left inverses under function composition, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps n to either ${\displaystyle 2n}$ or ${\displaystyle 2n+1}$ is a right inverse of the function ${\textstyle n\mapsto \left\lfloor {\frac {n}{2}}\right\rfloor ,}$ the floor function that maps n to ${\textstyle {\frac {n}{2}}}$ or ${\textstyle {\frac {n-1}{2}},}$ depending whether n is even or odd.

More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is surjective.

In category theory, right inverses are also called sections, and left inverses are called retractions.

An element is invertible under an operation if it has a left inverse and a right inverse.

In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if l and r are respectively a left inverse and a right inverse of x, then

${\displaystyle l=l*(x*r)=(l*x)*r=r.}$

The inverse of an invertible element is its unique left or right inverse.

If the operation is denoted as an addition, the inverse, or additive inverse, of an element x is denoted ${\displaystyle -x.}$ Otherwise, the inverse of x is generally denoted ${\displaystyle x^{-1},}$ or, in the case of a commutative multiplication ${\textstyle {\frac {1}{x}}.}$ When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in ${\displaystyle x^{*-1}.}$ The notation ${\displaystyle f^{\circ -1}}$ is not commonly used for function composition, since ${\textstyle {\frac {1}{f}}}$ can be used for the multiplicative inverse.

If x and y are invertible, and ${\displaystyle x*y}$ is defined, then ${\displaystyle x*y}$ is invertible, and its inverse is ${\displaystyle y^{-1}x^{-1}.}$

An invertible homomorphism is called an isomorphism. In category theory, an invertible morphism is also called an isomorphism.

Let ${\displaystyle S}$ be a unital magma, that is, a set with a binary operation ${\displaystyle *}$ and an identity element ${\displaystyle e\in S}$. If, for ${\displaystyle a,b\in S}$, we have ${\displaystyle a*b=e}$, then ${\displaystyle a}$ is called a left inverse of ${\displaystyle b}$ and ${\displaystyle b}$ is called a right inverse of ${\displaystyle a}$. If an element ${\displaystyle x}$ is both a left inverse and a right inverse of ${\displaystyle y}$, then ${\displaystyle x}$ is called a two-sided inverse, or simply an inverse, of ${\displaystyle y}$. An element with a two-sided inverse in ${\displaystyle S}$ is called invertible in ${\displaystyle S}$. An element with an inverse element only on one side is left invertible or right invertible.

Elements of a unital magma ${\displaystyle (S,*)}$ may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table

the elements 2 and 3 each have two two-sided inverses.

A unital magma in which all elements are invertible need not be a loop. For example, in the magma ${\displaystyle (S,*)}$ given by the Cayley table

every element has a unique two-sided inverse (namely itself), but ${\displaystyle (S,*)}$ is not a loop because the Cayley table is not a Latin square.

Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table

the only element with a two-sided inverse is the identity element 1.

If the operation ${\displaystyle *}$ is associative then if an element has both a left inverse and a right inverse, they are equal. In other words, 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 invertible elements is a group, called the group of units of ${\displaystyle S}$, and denoted by ${\displaystyle U(S)}$ or H₁.

The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity; that is, 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 as defined in this section. Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, that 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. This simple observation can be generalized using Green's relations: every idempotent e in an arbitrary semigroup is a left identity for R_e and right identity for L_e.[2] An intuitive description of this fact is that every pair of mutually inverse elements 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 in this section. Only elements in the Green class H₁ have an inverse from the unital magma perspective, whereas for any idempotent e, the elements of H_e 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 the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an inverse semigroup. Finally, an inverse semigroup with 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 is generally justified because in most applications (for example, all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity (see Generalized inverse).

A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (a°)° = a for all a in S; this endows S with a type ⟨2,1⟩ algebra. A semigroup endowed with such an operation is called a U-semigroup. Although it may seem that a° will be the inverse of a, this is not necessarily the case. In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classes of U-semigroups have been studied:[3]

• 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, and typically denoted by a*

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

All examples in this section involve associative operators.

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

A square matrix ${\displaystyle M}$ with entries in a field ${\displaystyle K}$ is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero. If the determinant of ${\displaystyle M}$ is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See invertible matrix for more.

More generally, a square matrix over a commutative ring ${\displaystyle R}$ is invertible if and only if its determinant is invertible in ${\displaystyle R}$.

Non-square matrices of full rank have several one-sided inverses:[4]

• For ${\displaystyle A:m\times n\mid m>n}$ we have left inverses; for example, ${\displaystyle \underbrace {\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}} _{A_{\text{left}}^{-1}}A=I_{n}}$
• For ${\displaystyle A:m\times n\mid m<n}$ we have right inverses; for example, ${\displaystyle A\underbrace {A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}} _{A_{\text{right}}^{-1}}=I_{m}}$

The left inverse can be used to determine the least norm solution of ${\displaystyle Ax=b}$, which is also the least squares formula for regression and is given by ${\displaystyle x=\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}b.}$

No rank deficient matrix has any (even one-sided) inverse. However, the Moore–Penrose inverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists.

As an example of matrix inverses, consider:

${\displaystyle A:2\times 3={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}}$

So, as m < n, we have a right inverse, ${\displaystyle A_{\text{right}}^{-1}=A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}.}$ By components it is computed as

${\displaystyle {\begin{aligned}AA^{\text{T}}&={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}{\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}={\begin{bmatrix}14&32\\32&77\end{bmatrix}}\\[3pt]\left(AA^{\text{T}}\right)^{-1}&={\begin{bmatrix}14&32\\32&77\end{bmatrix}}^{-1}={\frac {1}{54}}{\begin{bmatrix}77&-32\\-32&14\end{bmatrix}}\\[3pt]A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}&={\frac {1}{54}}{\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}{\begin{bmatrix}77&-32\\-32&14\end{bmatrix}}={\frac {1}{18}}{\begin{bmatrix}-17&8\\-2&2\\13&-4\end{bmatrix}}=A_{\text{right}}^{-1}\end{aligned}}}$

The left inverse doesn't exist, because

${\displaystyle A^{\text{T}}A={\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}{\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}={\begin{bmatrix}17&22&27\\22&29&36\\27&36&45\end{bmatrix}}}$

which is a singular matrix, and cannot be inverted.