Binary operation/Inverse element/Definition

Inverse element

Let a set ${}M$ with a binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

and a neutral element ${}e\in M$ be given. For an element ${}x\in M$, an element ${}y\in M$ is called inverse element (for ${}x$), if the equalities

${}x\circ y=e=y\circ x\,$

hold.