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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 (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 ...

addition
) and
reciprocation
reciprocation
(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 ...

multiplication
). 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 unital
magma 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\in S. If, for a,b\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''1.


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 a
semigroup 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 ''Re'' and right identity for ''Le''. 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
inverse 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 an
involution 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

Every
real 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 ...

addition
) 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 ...

multiplicative inverse
(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 ...

multiplication
) given by \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 ...

zero
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 (for
function 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 \circ f (resp. f \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 ...

identity function
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 ...

codomain
) 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 ...

bijection
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

A
square 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 ...

matrix multiplication
) 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 ...

determinant
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:MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse.
/ref> * For A:m\times n \mid m>n we have left inverses; for example, \underbrace_ A = I_n * For A:m\times n \mid m we have right inverses; for example, A \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 = \left(A^\textA\right)^A^\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 \times 3 = \begin 1 & 2 & 3 \\ 4 & 5 & 6 \end So, as ''m'' < ''n'', we have a right inverse, A^_\text = A^\text\left(AA^\text\right)^. By components it is computed as : \begin AA^\text &= \begin 1 & 2 & 3 \\ 4 & 5 & 6 \end\cdot \begin 1 & 4\\ 2 & 5\\ 3 & 6 \end = \begin 14 & 32\\ 32 & 77 \end \\ \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^\textA = \begin 1 & 4\\ 2 & 5\\ 3 & 6 \end \cdot \begin 1 & 2 & 3 \\ 4 & 5 & 6 \end = \begin 17 & 22 & 27 \\ 22 & 29 & 36\\ 27 & 36 & 45 \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