subsemigroup,, submonoid,, and subgroup
Let S be a semigroup, and let T be a subset of S.
T is a subsemigroup of S if T is closed under the operation of S; that it if xy∈T for all x,y∈T.
T is a submonoid of S if T is a subsemigroup, and T has an identity element.
T is a subgroup of S if T is a submonoid which is a group.
Note that submonoids and subgroups do not have to have the same identity element as S itself (indeed, S may not have an identity element). The identity element may be any idempotent element of S.
Let e∈S be an idempotent element. Then there is a maximal subsemigroup of S for which e is the identity:
eSe={exe∣x∈S}. |
In addition, there is a maximal subgroup for which e is the identity:
𝒰(eSe)={x∈eSe∣∃y∈eSestxy=yx=e}. |
Subgroups with different identity elements are disjoint. To see this, suppose that G and H are subgroups of a semigroup S with identity elements e and f respectively, and suppose x∈G∩H.
Then x has an inverse y∈G, and an inverse z∈H. We have:
e=xy=fxy=fe=zxe=zx=f. |
Thus intersecting subgroups have the same identity element.
Title | subsemigroup,, submonoid,, and subgroup |
---|---|
Canonical name | SubsemigroupSubmonoidAndSubgroup |
Date of creation | 2013-03-22 13:02:03 |
Last modified on | 2013-03-22 13:02:03 |
Owner | mclase (549) |
Last modified by | mclase (549) |
Numerical id | 5 |
Author | mclase (549) |
Entry type | Definition |
Classification | msc 20M99 |
Related topic | Semigroup |
Related topic | Subgroup |
Defines | subsemigroup |
Defines | submonoid |
Defines | subgroup |