Inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.
Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element such that x * e = x and e * y = y for all x and y for which the left-hand sides are defined.[1])
When the operation ∗ is
Inverses are commonly used in
The word 'inverse' is derived from
Definitions and basic properties
The concepts of inverse element and invertible element are commonly defined for
In this section, X is a
Associativity
A partial operation is
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.
Identity elements
Let be a possibly
An identity element, or simply an identity is an element e such that
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 is defined, then (This results immediately from the definition, by )
It follows that a total operation has at most one identity element, and if e and f are different identities, then 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.
Left and right inverses
If 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
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 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 or is a right inverse of the function the
More generally, a function has a left inverse for
In
Inverses
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
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 Otherwise, the inverse of x is generally denoted or, in the case of a
If x and y are invertible, and is defined, then is invertible, and its inverse is
An invertible homomorphism is called an isomorphism. In category theory, an invertible morphism is also called an isomorphism.
In groups
A
Thus, the inverse is a function from the group to itself that may also be considered as an operation of arity one. It is also an involution, since the inverse of the inverse of an element is the element itself.
A group may
For example, the
In monoids
A
The invertible elements in a monoid form a group under monoid operation.
A
If a monoid is not
For example, the set of the
Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the cancellation property (an element x has the cancellation property if implies and implies ). This extension of a monoid is allowed by Grothendieck group construction. This is the method that is commonly used for constructing integers from natural numbers, rational numbers from integers and, more generally, the field of fractions of an integral domain, and localizations of commutative rings.
In rings
A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers.
Under addition, a ring is an
Under multiplication, a ring is a
The additive identity 0 is never a unit, except when the ring is the zero ring, which has 0 as its unique element.
If 0 is the only non-unit, the ring is a field if the multiplication is commutative, or a division ring otherwise.
In a
A
Matrices
Matrix multiplication is commonly defined for matrices over a field, and straightforwardly extended to matrices over rings, rngs and semirings. However, in this section, only matrices over a commutative ring are considered, because of the use of the concept of rank and determinant.
If A is a m×n matrix (that is, a matrix with m rows and n columns), and B is a p×q matrix, the product AB is defined if n = p, and only in this case. An identity matrix, that is, an identity element for matrix multiplication is a square matrix (same number for rows and columns) whose entries of the main diagonal are all equal to 1, and all other entries are 0.
An
If R is a field, the determinant is invertible if and only if it is not zero. As the case of fields is more common, one see often invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings.
In the case of
A matrix has a left inverse if and only if its rank equals its number of columns. This left inverse is not unique except for square matrices where the left inverse equal the inverse matrix. Similarly, a right inverse exists if and only if the rank equals the number of rows; it is not unique in the case of a rectangular matrix, and equals the inverse matrix in the case of a square matrix.
Functions, homomorphisms and morphisms
In all the case, composition is
If and the composition is defined if and only if or, in the function and homomorphism cases, In the function and homomorphism cases, this means that the codomain of equals or is included in the domain of g. In the morphism case, this means that the codomain of equals the domain of g.
There is an identity for every object X (
A function is invertible if and only if it is a bijection. An invertible homomorphism or morphism is called an isomorphism. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an inverse function. In the other cases, one talks of inverse isomorphisms.
A function has a left inverse or a right inverse if and only it is
Generalizations
In a unital magma
Let be a unital magma, that is, a set with a binary operation and an identity element . If, for , we have , then is called a left inverse of and is called a right inverse of . If an element is both a left inverse and a right inverse of , then is called a two-sided inverse, or simply an inverse, of . An element with a two-sided inverse in is called invertible in . An element with an inverse element only on one side is left invertible or right invertible.
Elements of a unital magma may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
* | 1 | 2 | 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
* | 1 | 2 | 3 |
---|---|---|---|
1 | 1 | 2 | 3 |
2 | 2 | 1 | 2 |
3 | 3 | 2 | 1 |
every element has a unique two-sided inverse (namely itself), but 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
* | 1 | 2 | 3 | 4 | 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 is
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 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
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 He 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 ⟨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
Semirings
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
A square matrix with entries in a field 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 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 is invertible if and only if its determinant is invertible in .
Non-square matrices of
- For we have left inverses; for example,
- For we have right inverses; for example,
The left inverse can be used to determine the least norm solution of , which is also the least squares formula for regression and is given by
No
As an example of matrix inverses, consider:
So, as m < n, we have a right inverse, By components it is computed as
The left inverse doesn't exist, because
which is a
See also
- Division ring
- Latin square property
- Loop (algebra)
- Unit (ring theory)
Notes
- identity matrices as identity elements for matrix multiplication.
- ^ Howie, prop. 2.3.3, p. 51
- ^ Howie p. 102
- ^ "MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse".
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, ISBN 3-11-015248-7, p. 15 (def in unital magma) and p. 33 (def in semigroup)
- Howie, John M. (1995). Fundamentals of Semigroup Theory. 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, 24(1), December 1982, pp. 173–187
- Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377.