Inverse elements

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a 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.

The word 'inverse' is derived from Latin: that means 'turned upside down', 'overturned'.

Let ${\displaystyle S}$ be a set closed under a binary operation ${\displaystyle *}$ (i.e., a magma). If ${\displaystyle e}$ is an identity element of ${\displaystyle (S,*)}$ (i.e., S is a unital magma) and ${\displaystyle a*b=e}$, then ${\displaystyle a}$ is called a left inverse of ${\displaystyle b}$ and ${\displaystyle b}$ is called a right inverse of ${\displaystyle a}$. If an element ${\displaystyle x}$ is both a left inverse and a right inverse of ${\displaystyle y}$, then ${\displaystyle x}$ is called a two-sided inverse, or simply an inverse, of ${\displaystyle y}$. An element with a two-sided inverse in ${\displaystyle S}$ is called invertible in ${\displaystyle S}$. An element with an inverse element only on one side is left invertible, resp. right invertible. A unital magma in which all elements are invertible is called a loop. A loop whose binary operation satisfies the associative law is a group.

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