Inverse element
From Groupprops
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
Definition with symbols
Given a set with a binary operation
and a neutral element
for
, and given elements
and
we say that:
-
is a left inverse to
if
-
is a right inverse to
if
-
is an inverse or two-sided inverse to
if
(that is,
is both a left and a right inverse to
)
An element which possesses a (left/right) inverse is termed (left/right) invertible.
Facts
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 be an element of
with left inverse
and right inverse
. Then,
by associativity. The left side simplifies to
while the right side simplifies to
. Hence,
.
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