In abstract algebra, the idea of inverse element generalises the concepts of negation, in relation to addition, and 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.

Formal definitions

In an unital magma

Let S be a set with a binary operation * (i.e. a magma). If e is an identity element of (S,*) (i.e. S is an unital magma) and a*b=e, then a is called a left inverse of b and b is called a right inverse of a. If an element x is both a left inverse and a right inverse of y, then x is called a two-sided inverse, or simply an inverse, of y. An element with a two-sided inverse in S is called invertible in S. An element with an inverse element only on one side is left invertible, resp. right invertible. If all elements in S are invertible, S is called a loop.

Just like (S,*) can have several left identities or several right identities, it is possible for an element to have several left inverses or several right inverses (but note that their definition above uses a two-sided identity e). It can even have several left inverses and several right inverses.

If the operation * is associative then if an element has both a left inverse and a right inverse, they are equal. In other words, in a monoid every element has at most one inverse (as defined in this section). In a monoid, the set of (left and right) invertible elements is a group, called the group of units of S, and denoted by U(S) or H₁.

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 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 pseudoinverse. An element y is called (simply) an inverse of x if xyx = x and y = yxy. Every regular element has at least one inverse: if x = xzx then it is easy to verify that y = zxz is an inverse of x as defined in this section. 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. This simple observation can be generalized using Green's relations: every idempotent e in an arbitrary semigroup is a left identity for R_eand right identity for L_e. An intuitive description of this is fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.

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 H₁ have an inverse from the unital magma perspective, whereas for any idempotent e, the elements of H_e 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:

• 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. Inverse semigroups are exactly those semigroups that are both I-semigroups and *-semigroups. 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