subsemigroup,, submonoid,, and subgroup
Let be a semigroup, and let be a subset of .
is a submonoid of if is a subsemigroup, and has an identity element.
is a subgroup of if is a submonoid which is a group.
Note that submonoids and subgroups do not have to have the same identity element as itself (indeed, may not have an identity element). The identity element may be any idempotent element of .
In addition, there is a maximal subgroup for which is the identity:
Subgroups with different identity elements are disjoint. To see this, suppose that and are subgroups of a semigroup with identity elements and respectively, and suppose . Then has an inverse , and an inverse . We have:
Thus intersecting subgroups have the same identity element.
|Title||subsemigroup,, submonoid,, and subgroup|
|Date of creation||2013-03-22 13:02:03|
|Last modified on||2013-03-22 13:02:03|
|Last modified by||mclase (549)|