a characterization of groups
Theorem.
A non-empty semigroup is a group
if and only if
for every there is a unique such that .
Proof.
Suppose that is a non-empty semigroup, and for every there is a unique such that . For each , let denote the unique element of such that . Note that , so, by uniqueness, , and therefore .
For any , the element is idempotent (http://planetmath.org/Idempotency),
because .
As is nonempty, this means that has at least one idempotent element.
If is idempotent,
then , and so ,
and therefore ,
which means that .
So every idempotent is a left identity
,
and, by a symmetric
argument, a right identity.
Therefore, has at most one idempotent element.
Combined with the previous result,
this means that has exactly one idempotent element,
which we will denote by .
We have shown that is an identity
,
and that for each , so is a group.
Conversely, if is a group then clearly has a unique solution, namely . ∎
Note. Note that inverse semigroups do not in general
satisfy the hypothesis of this theorem:
in an inverse semigroup there is for each a unique such that and ,
but this need not be unique as a solution of alone.
Title | a characterization of groups |
---|---|
Canonical name | ACharacterizationOfGroups |
Date of creation | 2013-03-22 14:45:08 |
Last modified on | 2013-03-22 14:45:08 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 10 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 20A05 |
Related topic | Group |
Related topic | RegularSemigroup |
Related topic | AlternativeDefinitionOfGroup |