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#############################################################################
##
#W fpmon.gd GAP library Isabel Araujo
##
#H @(#)$Id: fpmon.gd,v 4.5 2002/04/15 10:04:40 sal 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 finitely presented monoids.
##
Revision.fpmon_gd :=
"@(#)$Id: fpmon.gd,v 4.5 2002/04/15 10:04:40 sal Exp $";
#############################################################################
##
#C IsElementOfFpMonoid(<elm>)
##
## returns true if <elm> is an element of a finitely presented monoid.
##
DeclareCategory( "IsElementOfFpMonoid",
IsMultiplicativeElementWithOne and IsAssociativeElement);
#############################################################################
##
#C IsElementOfFpMonoidCollection(<e>)
##
## Created now so that lists of things in the category IsElementOfFpMonoid
## are given the category CategoryCollections(IsElementOfFpMonoid)
## Otherwise these lists (and other collections) won't create the
## collections category. See CollectionsCategory in the manual.
##
DeclareCategoryCollections("IsElementOfFpMonoid");
#############################################################################
##
#A IsSubmonoidFpMonoid( <t> )
##
## true if <t> is a finitely presented monoid or a
## submonoid of a finitely presented monoid
## (generally speaking, such a semigroup can be constructed
## with `Monoid(<gens>)', where <gens> is a list of elements
## of a finitely presented monoid).
##
## A submonoid of a monoid has the same identity as the monoid.
##
DeclareSynonymAttr( "IsSubmonoidFpMonoid",
IsMonoid and IsElementOfFpMonoidCollection );
#############################################################################
##
#C IsElementOfFpMonoidFamily
##
DeclareCategoryFamily( "IsElementOfFpMonoid" );
#############################################################################
##
#F FactorFreeMonoidByRelations( <f>, <rels> )
##
## <f> is a free monoid and <rels> is a list of
## pairs of elements of <f>. Returns the fp monoid which
## is the quotient of <f> by the least congruence on <f> generated by
## the pairs in <rels>.
##
DeclareGlobalFunction("FactorFreeMonoidByRelations");
#############################################################################
##
#O ElementOfFpMonoid( <fam>, <word> )
##
## If <fam> is the elements family of a finitely presented monoid and <word>
## is a word in the free generators underlying this finitely presented
## monoid, this operation creates the element with the representative <word>
## in the free monoid.
##
DeclareOperation( "ElementOfFpMonoid",
[ IsElementOfFpMonoidFamily, IsAssocWordWithOne ] );
#############################################################################
##
#O FpMonoidOfElementOfFpMonoid( <elm> )
##
## returns the fp monoid to which <elm> belongs to
##
DeclareOperation( "FpMonoidOfElementOfFpMonoid",[IsElementOfFpMonoid]);
#############################################################################
##
#P IsFpMonoid(<m>)
##
## is a synonym for `IsSubmonoidFpMonoid(<m>)' and
## `IsWholeFamily(<m>)' (this is because a submonoid
## of a finitely presented monoid is not necessarily finitely presented).
##
DeclareSynonym( "IsFpMonoid",IsSubmonoidFpMonoid and IsWholeFamily);
#############################################################################
##
#A FreeGeneratorsOfFpMonoid( <m> )
##
## returns the underlying free generators corresponding to the
## generators of the finitely presented monoid <m>.
##
DeclareAttribute("FreeGeneratorsOfFpMonoid", IsFpMonoid);
#############################################################################
##
#A FreeMonoidOfFpMonoid( <m> )
##
## returns the underlying free monoid for the finitely presented
## monoid <m>, ie, the free monoid over which <m> is defined
## as a quotient
## (this is the free monoid generated by the free generators provided
## by `FreeGeneratorsOfFpMonoid(<m>)').
##
DeclareAttribute("FreeMonoidOfFpMonoid", IsFpMonoid);
############################################################################
##
#A RelationsOfFpMonoid(<m>)
##
## returns the relations of the finitely presented monoid <m> as
## pairs of words in the free generators provided by
## `FreeGeneratorsOfFpMonoid(<m>)'.
##
DeclareAttribute("RelationsOfFpMonoid",IsFpMonoid);
############################################################################
##
#A IsomorphismFpMonoid(<m>)
##
## for a monoid <m> returns an isomorphism from <m> to an fp monoid.
##
DeclareAttribute("IsomorphismFpMonoid",IsMonoid);
#############################################################################
##
#E
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