Schutzenberger group
In abstract algebra, in semigroup theory, a Schutzenberger group is a certain group associated with a Green H-class of a semigroup. The Schutzenberger groups associated with different H-classes are different. However, the groups associated with two different H-classes contained in the same D-class of a semigroup are isomorphic. Moreover, if the H-class itself were a group, the Schutzenberger group of the H-class would be isomorphic to the H-class. In fact, there are two Schutzenberger groups associated with a given H-class and each is antiisomorphic to the other.
The Schutzenberger group was discovered by Marcel-Paul Schützenberger in 1957 and the terminology was coined by A. H. Clifford.
Let S be a semigroup and let S1 be the semigroup obtained by adjoining an identity element 1 to S (if S already has an identity element, then S1 = S). Green's H-relation in S is defined as follows: If a and b are in S then
For a in S, the set of all b 's in S such that ' a H b ' is the Green H-class of S containing a, denoted by Ha.
Let H be an H-class of the semigroup S. Let T( H ) be the set of all elements t in S1 such that Ht is a subset of H itself. Each t in T( H ) defines a transformation, denoted by γt, of H by mapping h in H to ht in H. The set of all these transformations of H, denoted by Γ ( H ), is a group under composition of mappings (taking functions as right operators). The group Γ ( H ) is the Schutzenberger group associated with the H-class H.
If H is a maximal subgroup of a monoid M (a semigroup with identity), then H is an H-class, and it is naturally isomorphic to its own Schutzenberger group.
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