In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to 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.
Contents
- In a unital magma
- In a semigroup
- U semigroups
- Examples
- Real numbers
- Functions and partial functions
- Galois connections
- Matrices
- References
The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'.
In a unital magma
Let
Just like
If the operation
A left-invertible element is left-cancellative, and analogously for right and two-sided.
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, i.e. in a semigroup.
In a semigroup
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 (e.g. all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity.
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:
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 Penrose–Moore inverse.
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
Functions and partial functions
A function
Galois connections
The lower and upper adjoints in a (monotone) Galois connection, L and G are quasi-inverses of each other, i.e. 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
More generally, a square matrix over a commutative ring
Non-square matrices of full rank have several one-sided inverses:
The left inverse can be used to determine the least norm solution of
No rank deficient matrix has any (even one-sided) inverse. However, the Moore–Penrose pseudoinverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists.
As an example of matrix inverses, consider:
So, as m < n, we have a right inverse,
The left inverse doesn't exist, because
which is a singular matrix, and cannot be inverted.