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.
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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 twosided inverse, or simply an inverse, of y. An element with a twosided 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 twosided 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 H_{1}.
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