Editing Inverse element (section)

== Generalizations ==

=== In a unital magma ===
Let <math>S</math> be a unital [[Magma (algebra)|magma]], that is, a [[Set (mathematics)|set]] with a [[binary operation]] <math>*</math> and an [[identity element]] <math>e\in S</math>. If, for <math>a,b\in S</math>, we have <math>a*b=e</math>, then <math>a</math> is called a '''left inverse''' of <math>b</math> and <math>b</math> is called a '''right inverse''' of <math>a</math>. If an element <math>x</math> is both a left inverse and a right inverse of <math>y</math>, then <math>x</math> is called a '''two-sided inverse''', or simply an '''inverse''', of <math>y</math>. An element with a two-sided inverse in <math>S</math> is called '''invertible''' in <math>S</math>. An element with an inverse element only on one side is '''left invertible''' or '''right invertible'''.

Elements of a unital magma <math>(S,*)</math> may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
{|class="wikitable" style="text-align:center"
|-
!width=15|*
!width=15|1
!width=15|2
!width=15|3
|-
!1
|1
|2
|3
|-
!2
|2
|1
|1
|-
!3
|3
|1
|1
|}
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 (algebra)|loop]]. For example, in the magma <math>(S,*)</math> given by the [[Cayley table]]
{|class="wikitable" style="text-align:center"
|-
!width=15|*
!width=15|1
!width=15|2
!width=15|3
|-
!1
|1
|2
|3
|-
!2
|2
|1
|2
|-
!3
|3
|2
|1
|}
every element has a unique two-sided inverse (namely itself), but <math>(S,*)</math> 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
{|class="wikitable" style="text-align:center"
|-
!width=15|*
!width=15|1
!width=15|2
!width=15|3
!width=15|4
!width=15|5
|-
!1
|1
|2
|3
|4
|5
|-
!2
|2
|3
|1
|5
|4
|-
!3
|3
|4
|5
|1
|2
|-
!4
|4
|5
|2
|3
|1
|-
!5
|5
|1
|4
|2
|3
|}
the only element with a two-sided inverse is the identity element 1.

If the operation <math>*</math> 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 (mathematics)|group]], called the [[group of units]] of <math>S</math>, and denoted by <math>U(S)</math> or ''H''<sub>1</sub>.

=== In a semigroup ===
{{main|Regular semigroup}}
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 ''[[Generalized inverse|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 [[idempotent element|idempotent]]s, 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<sub>e</sub>'' and right identity for ''L<sub>e</sub>''.<ref>Howie, prop. 2.3.3, p. 51</ref> 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  [[Green's relations#The H and D relations|''H''<sub>1</sub>]] have an inverse from the unital magma perspective, whereas for any idempotent ''e'', the elements of ''H''<sub>e</sub> 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]]).

=== ''U''-semigroups ===

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 {{langle}}2,1{{rangle}} 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:<ref>Howie p. 102</ref>

* '''''I''-semigroups''', in which the interaction axiom is ''aa''°''a'' = ''a''
* '''[[Semigroup with involution|*-semigroups]]''', in which the interaction axiom is (''ab'')° = ''b''°''a''°. Such an operation is called an [[involution (mathematics)|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 semigroup]]s; 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 semigroup]]s. In contrast, a subclass of *-semigroups, the [[Semigroup with involution#Drazin|*-regular semigroup]]s (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''.

=== Semirings ===
{{main|Quasiregular element}}

=== Examples ===
All examples in this section involve associative operators.

==== Galois connections ====
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.

==== Generalized inverses of matrices <span class="anchor" id="matrices"></span><!-- [[Generalized inverse]] links here.  Please do not change. --> ====
A [[square matrix]] <math>M</math> with entries in a [[field (mathematics)|field]] <math>K</math> 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 <math>M</math> 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]] <math>R</math> is invertible [[if and only if]] its determinant is invertible in <math>R</math>.

Non-square matrices of [[full rank]] have several one-sided inverses:<ref>{{cite web| url = http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture33.htm| title = MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse.}}</ref>
* For <math>A:m\times n \mid m>n</math> we have left inverses; for example, <math>\underbrace{ \left(A^\text{T}A\right)^{-1} A^\text{T} }_{ A^{-1}_\text{left} } A = I_n</math>
* For <math>A:m\times n \mid m<n</math> we have right inverses; for example, <math>A \underbrace{ A^\text{T}\left(AA^\text{T}\right)^{-1} }_{ A^{-1}_\text{right} } = I_m</math>

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

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:

: <math>A:2 \times 3 =
  \begin{bmatrix}
    1 & 2 & 3 \\
    4 & 5 & 6
  \end{bmatrix}
</math>

So, as ''m'' < ''n'', we have a right inverse, <math>A^{-1}_\text{right} = A^\text{T} \left(AA^\text{T}\right)^{-1}.</math> By components it is computed as

: <math>\begin{align}
  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^{-1}_\text{right}
\end{align}</math>

The left inverse doesn't exist, because

: <math>
  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}
</math>

which is a [[singular matrix]], and cannot be inverted.