/usr/share/gap/lib/semitran.gd is in gap-libs 4r6p5-3.
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##
#W semitran.gd GAP library Isabel Araújo and Robert Arthur
##
##
#Y Copyright (C) 1997, Lehrstuhl D für 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
##
#############################################################################
##
#P IsTransformationSemigroup( <obj> )
#P IsTransformationMonoid( <obj> )
##
## <#GAPDoc Label="IsTransformationSemigroup">
## <ManSection>
## <Prop Name="IsTransformationSemigroup" Arg='obj'/>
## <Prop Name="IsTransformationMonoid" Arg='obj'/>
##
## <Description>
## 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.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr("IsTransformationSemigroup", IsSemigroup and
IsTransformationCollection);
DeclareProperty("IsTransformationMonoid", IsTransformationSemigroup);
#############################################################################
##
#P IsFullTransformationSemigroup(<obj>)
##
## <#GAPDoc Label="IsFullTransformationSemigroup">
## <ManSection>
## <Prop Name="IsFullTransformationSemigroup" Arg='obj'/>
##
## <Description>
## checks whether <A>obj</A> is a full transformation semigroup.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty("IsFullTransformationSemigroup", IsSemigroup);
#############################################################################
##
#F FullTransformationSemigroup(<degree>)
##
## <#GAPDoc Label="FullTransformationSemigroup">
## <ManSection>
## <Func Name="FullTransformationSemigroup" Arg='degree'/>
##
## <Description>
## Returns the full transformation semigroup of degree <A>degree</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("FullTransformationSemigroup");
#############################################################################
##
#A DegreeOfTransformationSemigroup( <S> )
##
## <#GAPDoc Label="DegreeOfTransformationSemigroup">
## <ManSection>
## <Attr Name="DegreeOfTransformationSemigroup" Arg='S'/>
##
## <Description>
## The number of points the semigroup <A>S</A> acts on.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("DegreeOfTransformationSemigroup",
IsTransformationSemigroup);
############################################################################
##
#A IsomorphismTransformationSemigroup(<S>)
#O HomomorphismTransformationSemigroup(<S>,<r>)
##
## <#GAPDoc Label="IsomorphismTransformationSemigroup">
## <ManSection>
## <Attr Name="IsomorphismTransformationSemigroup" Arg='S'/>
## <Oper Name="HomomorphismTransformationSemigroup" Arg='S, r'/>
##
## <Description>
## <Ref Func="IsomorphismTransformationSemigroup"/> is a generic attribute
## which is a transformation semigroup isomorphic to <A>S</A> (if such can
## be computed).
## In the case of an fp-semigroup, a Todd-Coxeter approach
## will be attempted. For a semigroup of endomorphisms of a finite
## domain of <M>n</M> elements, it will be to a semigroup of transformations
## of <M>\{ 1, 2, \ldots, n \}</M>. Otherwise, it will be the right regular
## representation on <A>S</A> or <M><A>S</A>^1</M> if <A>S</A> has no
## multiplicative neutral element,
## see <Ref Func="MultiplicativeNeutralElement"/>.
## <P/>
## <Ref Func="HomomorphismTransformationSemigroup"/>
## finds a representation of <A>S</A> as transformations of the set of
## equivalence classes of the right congruence <A>r</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("IsomorphismTransformationSemigroup",
IsSemigroup);
DeclareOperation("HomomorphismTransformationSemigroup",
[IsSemigroup,IsRightMagmaCongruence]);
#############################################################################
##
#E
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