a characterization of groups
A non-empty semigroup is a group if and only if for every there is a unique such that .
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.
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|
|Date of creation||2013-03-22 14:45:08|
|Last modified on||2013-03-22 14:45:08|
|Last modified by||yark (2760)|