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#############################################################################
##
#W semitran.gd GAP library Isabel Araujo and Robert Arthur
##
#H @(#)$Id: semitran.gd,v 4.8.2.1 2004/02/20 16:40:38 gap Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the declarations for basics of transformation semigroup
Revision.semitran_gd :=
"@(#)$Id: semitran.gd,v 4.8.2.1 2004/02/20 16:40:38 gap Exp $";
#############################################################################
##
#P IsTransformationSemigroup( <obj> )
#P IsTransformationMonoid( <obj> )
##
## A transformation semigroup (resp. monoid) is a subsemigroup
## (resp. submonoid) of the full transformation monoid.
## Note that for a transformation semigroup to be a transformation monoid
## we necessarily require the identity transformation to be an element.
##
DeclareSynonymAttr("IsTransformationSemigroup", IsSemigroup and
IsTransformationCollection);
DeclareProperty("IsTransformationMonoid", IsTransformationSemigroup);
#############################################################################
##
#P IsFullTransformationSemigroup(<obj>)
##
DeclareProperty("IsFullTransformationSemigroup", IsSemigroup);
#############################################################################
##
#F FullTransformationSemigroup(<degree>)
##
## Returns the full transformation semigroup of degree <degree>.
##
DeclareGlobalFunction("FullTransformationSemigroup");
#############################################################################
##
#A DegreeOfTransformationSemigroup( <S> )
##
## The number of points the semigroup acts on.
DeclareAttribute("DegreeOfTransformationSemigroup",
IsTransformationSemigroup);
############################################################################
##
#A IsomorphismTransformationSemigroup(<S>)
#O HomomorphismTransformationSemigroup(<S>,<r>)
##
##
## IsomorphismTransformationSemigroup is a generic attribute which
## is a transformation semigroup isomorphic to <S> (if such can
## be computed). In the case of an fp- semigroup, a todd-coxeter
## will be attempted. For a semigroup of endomorphisms of a finite
## domain of <n> elements, it will be to a semigroup of transformations
## of $\{1, \ldots, n\}$. Otherwise, it will be the right regular
## representation on $S$ or $S^1$ if $S$ has no
## MultiplicativeNeutralElement.
##
## HomomorphismTransformationSemigroup
## finds a representation of <S> as transformations of the set of
## equivalence classes of the right congruence <r>.
##
DeclareAttribute("IsomorphismTransformationSemigroup",
IsSemigroup);
DeclareOperation("HomomorphismTransformationSemigroup",
[IsSemigroup,IsRightMagmaCongruence]);
#############################################################################
##
#E
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