4.10 H Classes for Transformation Monoids

In addition to the usual components of an H class record, the record representing the H class hClass of s in a transformation monoid can have the following components. They are created by the function SchutzenbergerGroup (see SchutzenbergerGroup) which is called whenever the size, the list of elements of hClass, or a membership test in hClass is asked for.

schutzenbergerGroup:

set to the Schaccent127utzenberger group of hClass as a permutation group on the set of images of SchutzenbergerGroup for Transformation Monoids).

R:

the R class of hClass.representative.

L:

the L class of hClass.representative. The following functions have a special implementation in terms of these components.

Size( hClass )

returns the size of the H class hClass. This function calls SchutzenbergerGroup and determines the size of hClass as the size of the resulting group.

Elements( hClass )

returns the set of elements of the H class hClass. This function calls SchutzenbergerGroup and determines the set of elements of hClass as the set of elements of the resulting group multiplied by the representative of hClass.

x in hClass

returns true if x is an element of the H class hClass and false otherwise. This function calls SchutzenbergerGroup and tests whether the quotient of the representative of hClass and x (see PermLeftQuoTrans) is in the resulting group.

   
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