4 Transformation Monoids

A transformation monoid is a monoid of transformations of n points (see chapter Transformations). These monoids occur for example in the theory of finite state automata and as the results of enumerations of finitely presented monoids. In the theory of semigroups and monoids they play to some extend the role that is taken by permutation groups in group theory. In fact, there are close relations between permutation groups and transformation monoids. One of these relations is manifested by the Schaccent127utzenberger group of an element of a transformation monoid, which is represented as a permutation group rather than a group of transformations. Another relation which is used by most functions that deal with transformation monoids is the fact that a transformation monoid can be efficiently described in terms of several permutation groups (for details see~LPRR1 and~LPRR2).

This chapter describes the functions that deal with transformation monoids.

The chapter starts with the description of the function that tests whether or not a given object is a transformation monoid (see IsTransMonoid). Then there is the function that determines the degree of a transformation monoid (see Degree of a Transformation Monoid).

There are a function to construct the full transformation monoid of degree n (see FullTransMonoid) and a function to construct the monoid of all partial transformations of degree n (see PartialTransMonoid).

Then there are a function that determines all images of a transformation monoid (see ImagesTransMonoid) and a function that determines all kernels of a transformation monoid (see KernelsTransMonoid).

Because each transformation monoid is a domain all set theoretic Set Functions for Transformation Monoids). Also because a transformation monoid is after all a monoid all monoid functions can be applied to it Monoid Functions for Transformation Monoids).

Next the functions that determine Green classes in transformation monoids D Classes for Transformation Monoids).

Finally, there is a section about how a transformation monoid is displayed (see Display a Transformation Monoid). The last section in this chapter describes how transformation monoids are represented as records in GAP (see Transformation Monoid Records).

The functions described here are in the file "monotran.g".


  1. IsTransMonoid
  2. Degree of a Transformation Monoid
  3. FullTransMonoid
  4. PartialTransMonoid
  5. ImagesTransMonoid
  6. KernelsTransMonoid
  7. Set Functions for Transformation Monoids
  8. Monoid Functions for Transformation Monoids
  9. SchutzenbergerGroup for Transformation Monoids
  10. H Classes for Transformation Monoids
  11. R Classes for Transformation Monoids
  12. L Classes for Transformation Monoids
  13. D Classes for Transformation Monoids
  14. Display a Transformation Monoid
  15. Transformation Monoid Records
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Version 2.4 (May 1998)