4.11 R Classes for Transformation Monoids

In addition to the usual components of an R class record, the record representing the R class rClass 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 rClass, or a membership test in rClass is asked for.

schutzenbergerGroup:

set to the Schaccent127utzenberger group of rClass as a permutation group on the set of images of SchutzenbergerGroup for Transformation Monoids). images:
is the list of different image sets occurring in the R class rClass. The first entry in this list is the image set of rClass.representative. rMults:
is a list of permutations such that the product of the representative of rClass and the inverse of the ith entry in the list yields an element of rClass whose image set is the ith entry in the list rClass.images.

The following functions have a special implementation in terms of these components.

Size( rClass )

returns the size of the R class rClass. This function calls SchutzenbergerGroup and determines the size of rClass as the size of the resulting group times the length of the list rClass.images.

Elements( rClass )

returns the set of elements of the R class rClass. This function calls SchutzenbergerGroup and determines the set of elements of rClass as the set of elements of the resulting group multiplied by the representative of rClass and each single permutation in the list rClass.rMults.

x in rClass

returns true if x is an element of the R class rClass and false otherwise. This function calls SchutzenbergerGroup and tests whether the quotient of the representative of rClass and x * rClass.rMults[i] (see PermLeftQuoTrans) is in the resulting group where i is the position of the image set of x in rClass.images.

HClasses( rClass )

returns the list of H classes contained in the R class rClass.

   
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