In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the inverse element or simply the inverse. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. Inverses are commonly used in groupswhere every element is invertible, and ringswhere invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism. The word 'inverse' is derived from that means 'turned upside down', 'overturned'. This may take its origin from the case of fractions, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of is ). Definitions and basic properties The concepts of inverse element and invertible element are commonly defined for binary operations that are everywhere defined (that is, the operation is defined for any two elements of its domain). However, these concepts are commonly used with partial operations, that is operations that are not defined everywhere. Common examples are matrix multiplication, function composition and composition of morphisms in a category. 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. In this section, is a set (possibly a proper class) on which a partial operation (possibly total) is defined, which is denoted with Associativity A partial operation is associative if for every in 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