In

_{1}.

_{e}'' and right identity for ''L_{e}''. 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 ''H''_{1} 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

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

/ref> * For $A:m\backslash times\; n\; \backslash mid\; m>n$ we have left inverses; for example, $\backslash underbrace\_\; A\; =\; I\_n$ * For $A:m\backslash times\; n\; \backslash mid\; mmath>\; we\; have\; right\; inverses;\; for\; example,$ A\; \backslash underbrace\_\; =\; I\_m$The\; left\; inverse\; can\; be\; used\; to\; determine\; the\; least\; norm\; solution\; of$ Ax\; =\; b$,\; which\; is\; also\; the$

abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, the idea of an inverse element generalises the concepts of negation (sign reversal) (in relation to addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

) and (in relation to multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

). The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

.
The word 'inverse' is derived from la, inversus that means 'turned upside down', 'overturned'.
Formal definitions

In a unital magma

Let $S$ be a unitalmagma
Magma () is the molten or semi-molten natural material from which all igneous rock
Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...

, that is, a set with a binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$*$ and an identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

$e\backslash in\; S$. If, for $a,b\backslash in\; S$, we have $a*b=e$, then $a$ is called a left inverse of $b$ and $b$ 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 called a two-sided inverse, or simply an inverse, of $y$. An element with a two-sided inverse in $S$ is called invertible in $S$. An element with an inverse element only on one side is left invertible or right invertible.
Elements of a unital magma $(S,*)$ 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 $(S,*)$ given by the Cayley table Named after the 19th century British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity a ...

every element has a unique two-sided inverse (namely itself), but $(S,*)$ is not a loop because the Cayley table is not a Latin square
In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 La ...

.
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 $*$ is associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

then if an element has both a left inverse and a right inverse, they are equal. In other words, in a monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

(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
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

, called the group of units
In the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...

of $S$, and denoted by $U(S)$ or ''H''In a 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 asemigroup
In mathematics, a semigroup is an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

.
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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

''. 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
Idempotence (, ) is the property of certain operations in mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

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 relationsIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

: every idempotent ''e'' in an arbitrary semigroup is a left identity for ''Rinverse semigroupIn group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', ...

. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an absorbing elementIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

).
''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 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: * ''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 aninvolution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...

, 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 semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a Class (set theory), class of semigroups satisfying additional property (philosophy), properties or conditions. Thus the ...

s. 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 inverseIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

. 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''.
Rings and semirings

Examples

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

Everyreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

$x$ has an additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

(that is, an inverse with respect to addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

) given by $-x$. Every nonzero real number $x$ has a multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

(that is, an inverse with respect to multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

) given by $\backslash frac\; 1$ (or $x^$). By contrast, zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

has no multiplicative inverse, but it has a unique quasi-inverse, "$0$" itself.
Functions and partial functions

A function $g$ is the left (resp. right) inverse of a function $f$ (forfunction composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

), if and only if $g\; \backslash circ\; f$ (resp. $f\; \backslash circ\; g$) is the identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

on the domain
Domain may refer to:
Mathematics
*Domain of a function
In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...

(resp. codomain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) of $f$. The inverse of a function $f$ is often written $f^$, but this notation is sometimes ambiguous. Only bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s have two-sided inverses, but ''any'' function has a quasi-inverse; that is, the full transformation monoidIn algebra, a transformation semigroup (or composition semigroup) is a collection of Transformation (function), transformations (function (mathematics), functions from a set to itself) that is closure (mathematics), closed under function composition. ...

is regular. The monoid of partial functions is also regular, whereas the monoid of injective partial transformations is the prototypical inverse semigroup.
Galois connections

The lower and upper adjoints in a (monotone)Galois connection In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...

, ''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.
Matrices

Asquare matrix
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$M$ with entries in a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

$K$ is invertible (in the set of all square matrices of the same size, under matrix multiplication
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

) if and only if its determinant
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

is different from zero. If the determinant of $M$ 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
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

for more.
More generally, a square matrix over a commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...

$R$ is invertible if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

its determinant is invertible in $R$.
Non-square matrices of full rank have several one-sided inverses:/ref> * For $A:m\backslash times\; n\; \backslash mid\; m>n$ we have left inverses; for example, $\backslash underbrace\_\; A\; =\; I\_n$ * For $A:m\backslash times\; n\; \backslash mid\; mmath>\; we\; have\; right\; inverses;\; for\; example,$ A\; \backslash underbrace\_\; =\; I\_m$The\; left\; inverse\; can\; be\; used\; to\; determine\; the\; least\; norm\; solution\; of$ Ax\; =\; b$,\; which\; is\; also\; the$

least squares
The method of least squares is a standard approach in regression analysis
In ing, regression analysis is a set of statistical processes for the relationships between a (often called the 'outcome' or 'response' variable) and one or more s ...

formula for regression and is given by $x\; =\; \backslash left(A^\backslash textA\backslash right)^A^\backslash textb.$
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:
: $A:2\; \backslash times\; 3\; =\; \backslash begin\; 1\; \&\; 2\; \&\; 3\; \backslash \backslash \; 4\; \&\; 5\; \&\; 6\; \backslash end$
So, as ''m'' < ''n'', we have a right inverse, $A^\_\backslash text\; =\; A^\backslash text\backslash left(AA^\backslash text\backslash right)^.$ By components it is computed as
: $\backslash begin\; AA^\backslash text\; \&=\; \backslash begin\; 1\; \&\; 2\; \&\; 3\; \backslash \backslash \; 4\; \&\; 5\; \&\; 6\; \backslash end\backslash cdot\; \backslash begin\; 1\; \&\; 4\backslash \backslash \; 2\; \&\; 5\backslash \backslash \; 3\; \&\; 6\; \backslash end\; =\; \backslash begin\; 14\; \&\; 32\backslash \backslash \; 32\; \&\; 77\; \backslash end\; \backslash \backslash $ \left(AA^\text\right)^ &= \begin
14 & 32\\
32 & 77
\end^ = \frac
\begin
77 & -32\\
-32 & 14
\end \\ A^\text\left(AA^\text\right)^ &= \frac
\begin
1 & 4\\
2 & 5\\
3 & 6
\end\cdot
\begin
77 & -32\\
-32 & 14
\end = \frac
\begin
-17 & 8\\
-2 & 2\\
13 & -4
\end = A^_\text
\end
The left inverse doesn't exist, because
: $A^\backslash textA\; =\; \backslash begin\; 1\; \&\; 4\backslash \backslash \; 2\; \&\; 5\backslash \backslash \; 3\; \&\; 6\; \backslash end\; \backslash cdot\; \backslash begin\; 1\; \&\; 2\; \&\; 3\; \backslash \backslash \; 4\; \&\; 5\; \&\; 6\; \backslash end\; =\; \backslash begin\; 17\; \&\; 22\; \&\; 27\; \backslash \backslash \; 22\; \&\; 29\; \&\; 36\backslash \backslash \; 27\; \&\; 36\; \&\; 45\; \backslash end$
which is a singular matrix
Singular may refer to:
* Singular, the grammatical number
In linguistics, grammatical number is a grammatical category
A grammatical category or grammatical feature is a property of items within the grammar
In linguistics
Linguist ...

, and cannot be inverted.
See also

*Division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...

*
*Loop (algebra)
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group (mathematics), group in the sense that "division (mathematics), division" is always possible. Quasigroups differ from groups mainly in that the ...

*Unit (ring theory)
In the branch of abstract algebra known as ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with nu ...

Notes

References

* M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, , p. 15 (def in unital magma) and p. 33 (def in semigroup) *{{cite book, last= Howie, first= John M., title=Fundamentals of Semigroup Theory, year=1995, publisher=Clarendon Press
Oxford University Press (OUP) is the university press 200px, The Pitt Building in Cambridge, which used to be the headquarters of Cambridge University Press, and now serves as a conference centre for the Press.
A university press is an academic ...

, isbn=0-19-851194-9 contains all 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
Semigroup Forum (print , electronic ) is a mathematics research journal published by Springer Science+Business Media, Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigr ...

, 24(1), December 1982, pp. 173–187
*Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum
Semigroup Forum (print , electronic ) is a mathematics research journal published by Springer Science+Business Media, Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigr ...

, 16(1978), 369–377.
Algebra
Abstract algebra
*Inverse element