Schützenberger group


In semigroup theory, a Schützenberger group is a group associated with a Green of a semigroup. The Schützenberger groups associated with different are distinct, but the groups associated with two different contained in the same of a semigroup are isomorphic. Moreover, if the itself were a group, the Schützenberger group of the would be isomorphic to the. In fact, there are two Schützenberger groups associated with a given, with each being antiisomorphic to the other.
The Schützenberger group was discovered by Marcel-Paul Schützenberger in 1957 and the terminology was coined by A. H. Clifford.

The Schützenberger group

Let S be a semigroup and let S1 be the semigroup obtained by adjoining an identity element 1 to S. Green's 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 of S containing a, denoted by Ha.
Let H be an of the semigroup S. Let T be the set of all elements t in S1 such that Ht is a subset of H itself. Each t in T 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, is a group under composition of mappings. The group Γ is the Schützenberger group associated with the H.

Examples

If H is a maximal subgroup of a monoid M, then H is an, and it is naturally isomorphic to its own Schützenberger group.
In general, one has that the cardinality of H and its Schützenberger group coincide for any H.

Applications

It is known that a monoid with finitely many left and right ideals is finitely presented if and only if all of its Schützenberger groups are finitely presented. Similarly such a monoid is residually finite if and only if all of its Schützenberger groups are residually finite.