Inverse element

''This is the default notion of inverse element. There is another, more general notion of inverse element in a semigroup, which does not depend on existence of a neutral element''

Definition with symbols
Given a set $$S$$ with a binary operation $$*$$ and a neutral element $$e$$ for $$*$$, and given elements $$a$$ and $$b$$ we say that:


 * $$b$$ is a left inverse to $$a$$ if $$b * a = e$$
 * $$b$$ is a right inverse to $$a$$ if $$a * b = e$$
 * $$b$$ is an inverse or two-sided inverse to $$a$$ if $$a * b = b * a = e$$ (that is, $$b$$ is both a left and a right inverse to $$a$$)

An element which possesses a (left/right) inverse is termed (left/right) invertible.

Equality of left and right inverses
If $$*$$ is an associative binary operation, and an element has both a left and a right inverse with respect to $$*$$, then the left and right inverse are equal.

To prove this, let $$a$$ be an element of $$S$$ with left inverse $$b$$ and right inverse $$c$$. Then, $$(b * a) * c = b * (a * c)$$ by associativity. The left side simplifies to $$e * c = c$$ while the right side simplifies to $$b * e = b$$. Hence, $$b = c$$.

Some easy corollaries:


 * If an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse
 * If an element has a right inverse, it can have at most one left inverse; moreover, if the left inverse exists, it must be equal to the right inverse, and is thus a two-sided inverse