PlanetMath: subsemigroup,, submonoid,, and subgroup

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

subsemigroup, submonoid, and subgroup

(Definition)

Let be a semigroup, and let be a subset of .

is a subsemigroup of if is closed under the operation of ; that it if $xy \in T$ for all $x, y \in T$ .

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 .

Let $e \in S$ be an idempotent element. Then there is a maximal subsemigroup of for which is the identity:

$\displaystyle eSe = \{ exe \mid x \in S \}.$

In addition, there is a maximal subgroup for which

is the identity:

$\displaystyle \mathcal{U}(eSe) = \{x \in eSe \mid \exists y \in eSe \;$ st $\displaystyle \; xy=yx=e \}.$

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 $x \in G \cap H$ . Then has an inverse $y \in G$ , and an inverse $z \in H$ . We have:

$\displaystyle e = xy = fxy = fe = zxe = zx = f.$

Thus intersecting subgroups have the same identity element.

"subsemigroup, submonoid, and subgroup" is owned by mclase.

(view preamble)