4.12 L Classes for Transformation Monoids

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

schutzenbergerGroup:

set to the Schaccent127utzenberger group of lClass as a permutation group on the set of images of SchutzenbergerGroup for Transformation Monoids). kernels:
is the list of different kernels occurring in the L class lClass. The first entry in this list is the kernel of rClass.representative. lMults:
is a list of binary relations such that the product of the inverse of the ith entry in the list and the representative of rClass yields an element of rClass whose kernel is the ith entry in the list rClass.kernels.

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

Size( lClass )

returns the size of the L class lClass. This function calls SchutzenbergerGroup and determines the size of lClass as the size of the resulting group times the length of the list lClass.kernels.

Elements( lClass )

returns the set of elements of the L class lClass. This function calls SchutzenbergerGroup and determines the set of elements of lClass as the set of elements of the resulting group premultiplied by the representative of lClass and each single binary relation in the list lClass.lMults.

x in lClass

returns true if x is an element of the L class lClass and false otherwise. This function calls SchutzenbergerGroup and tests whether the quotient of the representative of lClass and lClass.lMults[i] * x (see PermLeftQuoTrans) is in the resulting group where i is the position of the kernel of x in lClass.kernels.

HClasses( lClass )

returns the list of H classes contained in the L class lClass.

   
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