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 is denoted Otherwise, the inverse of is generally denoted or, in the case of a commutative multiplication When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in The notation is not commonly used for function composition, since can be used for the multiplicative inverse. If and 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 group is a set with an associative operation that has an identity element, and for which every element has an inverse. 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 act on a set as transformations of this set. In this case, the inverse of a group element defines a transformation that is the inverse of the transformation defined by that is, the transformation that "undoes" the transformation defined by For example, the Rubik's cube group represents the finite sequences of elementary moves. The inverse of such a sequence is obtained by applying the inverse of each move in the reverse order. In monoids A monoid is a set with an associative operation that has an identity element. The invertible elements in a monoid form a group under monoid operation. A ring is a monoid for ring multiplication. In this case, the invertible elements are also called units and form the group of units of the ring. If a monoid is not commutative, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible). For example, the set of the functions from a set to itself is a monoid under function composition. In this monoid, the invertible elements are the bijective functions; the elements that have left inverses are the injective functions, and those that have right inverses are the surjective functions. Given a monoid, one may want extend it by adding inverse