This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism.
The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'.
It follows that the common definitions of associativity and identity element must be extended to partial operations; this is the object of the first subsections.
Examples of non-total associative operations are multiplication of matrices of arbitrary size, and function composition.
Left and right inverses do not always exist, even when the operation is total and associative.
This lack of inverses is the main motivation for extending the natural numbers into the integers.
In the common case where the operation is associative, the left and right inverse of an element are equal and unique.
In category theory, an invertible morphism is also called an isomorphism.
Thus, the inverse is a function from the group to itself that may also be considered as an operation of arity one.
For example, the Rubik's cube group represents the finite sequences of elementary moves.
A monoid is a set with an associative operation that has an identity element.
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
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.
This is, for example, the case of the linear functions from an infinite-dimensional vector space to itself.
However, in this section, only matrices over a commutative ring are considered, because of the use of the concept of rank and determinant.
A matrix has a left inverse if and only if its rank equals its number of columns.
equals or is included in the domain of g. In the morphism case, this means that the codomain of
For example, the converse is true for vector spaces but not for modules over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a split epimorphism or a split monomorphism.
For example, in the magma given by the Cayley table the elements 2 and 3 each have two two-sided inverses.
In other words, in a monoid (an associative unital magma) every element has at most one inverse (as defined in this section).
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.
An element y is called (simply) an inverse of x if xyx = x and y = yxy.
Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, 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.
Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid.
Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse.
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.
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.
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.
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.
By components it is computed as The left inverse doesn't exist, because which is a singular matrix, and cannot be inverted.