In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, in semigroup theory, a Schutzenberger group is a certain group associated with a Green H-class of a semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

. 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

Antiisomorphic

In modern algebra, an antiisomorphism between structured sets A and B is an isomorphism from A to the opposite of B...

 to the other.

The Schutzenberger group was discovered by Marcel-Paul Schützenberger

Marcel-Paul Schützenberger

Marcel-Paul "Marco" Schützenberger was a French mathematician and Doctor of Medicine. His work had impact across the fields of formal language, combinatorics, and information theory...

 in 1957 and the terminology was coined by A. H. Clifford.

The Schutzenberger group

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

a H b ⇔ there are u, v, x, y in S¹ such that ua = vb and ax = by.

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 H_a.

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 γ_t, 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

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

 of mappings (taking functions as right oprators). The group Γ( H ) is the Schutzenberger group associated with the H-class H.

Examples

If H is a maximal subgroup of a monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

 M (a semigroup with identity), then H is an H-class, and it is naturally isomorphic to its own Schutzenberger group.

In general, one has that the cardinality of H and its Schutzenberger group coincide for any H-class H.

Applications

It is known that a monoid with finitely many left and right ideals is finitely presented (or just finitely generated) if and only if all of its Schutzenberger groups are finitely presented (respectively, finitely generated). Similarly such a monoid is residually finite if and only if all of its Schutzenberger groups are residually finite.