subsemigroup,, submonoid,, and subgroup

Let S be a semigroupPlanetmathPlanetmath, 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 xyT for all x,yT.

T is a submonoid of S if T is a subsemigroup, and T has an identity elementMathworldPlanetmath.

T is a subgroupMathworldPlanetmath 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 eS be an idempotent element. Then there is a maximal subsemigroup of S for which e is the identityPlanetmathPlanetmathPlanetmath:


In addition, there is a maximal subgroup for which e is the identity:


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 xGH. Then x has an inverseMathworldPlanetmath yG, and an inverse zH. We have:


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