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#W  semiquo.gd           GAP library          Andrew Solomon and Isabel Araújo
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#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
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##  This file contains the declarations for quotient semigroups.
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##  <#GAPDoc Label="[1]{semiquo}">
##  For a semigroup <M>S</M>,
##  elements of a quotient semigroup are equivalence classes of 
##  elements of the <Ref Func="QuotientSemigroupPreimage"/> value
##  under the congruence given by the value of
##  <Ref Func="QuotientSemigroupCongruence"/>.
##  <P/>
##  It is probably most useful for calculating the elements of 
##  the equivalence classes by using <Ref Func="Elements"/> or by looking at
##  the images of elements of <Ref Func="QuotientSemigroupPreimage"/> under
##  the map returned by <Ref Func="QuotientSemigroupHomomorphism"/>,
##  which maps the <Ref Func="QuotientSemigroupPreimage"/> value to <A>S</A>.
##  <P/>
##  For intensive computations in a quotient semigroup, it is probably
##  worthwhile finding another representation as the equality test 
##  could involve enumeration of the elements of the congruence classes
##  being compared.
##  <#/GAPDoc>
##


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#C  IsQuotientSemigroup( <S> )
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##  <#GAPDoc Label="IsQuotientSemigroup">
##  <ManSection>
##  <Filt Name="IsQuotientSemigroup" Arg='S' Type='Category'/>
##
##  <Description>
##  is the category of semigroups constructed from another semigroup 
##  and a congruence on it.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategory("IsQuotientSemigroup", IsSemigroup);

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#F  HomomorphismQuotientSemigroup(<cong>)
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##  <#GAPDoc Label="HomomorphismQuotientSemigroup">
##  <ManSection>
##  <Func Name="HomomorphismQuotientSemigroup" Arg='cong'/>
##
##  <Description>
##  for a congruence <A>cong</A> and a semigroup <A>S</A>. 
##  Returns the homomorphism from <A>S</A> to the quotient of <A>S</A> 
##  by <A>cong</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction("HomomorphismQuotientSemigroup");

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#A  QuotientSemigroupPreimage(<S>)
#A  QuotientSemigroupCongruence(<S>)
#A  QuotientSemigroupHomomorphism(<S>)
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##  <#GAPDoc Label="QuotientSemigroupPreimage">
##  <ManSection>
##  <Attr Name="QuotientSemigroupPreimage" Arg='S'/>
##  <Attr Name="QuotientSemigroupCongruence" Arg='S'/>
##  <Attr Name="QuotientSemigroupHomomorphism" Arg='S'/>
##
##  <Description>
##  for a quotient semigroup <A>S</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
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DeclareAttribute("QuotientSemigroupPreimage", IsQuotientSemigroup);
DeclareAttribute("QuotientSemigroupCongruence", IsQuotientSemigroup);
DeclareAttribute("QuotientSemigroupHomomorphism", IsQuotientSemigroup);


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