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#############################################################################
##
#W monoid.gd GAP library Thomas Breuer
##
#H @(#)$Id: monoid.gd,v 4.20 2002/04/15 10:05:04 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 declaration of operations for monoids.
##
Revision.monoid_gd :=
"@(#)$Id: monoid.gd,v 4.20 2002/04/15 10:05:04 sal Exp $";
#############################################################################
##
#P IsMonoid( <D> )
##
## A *monoid* is a magma-with-one (see~"Magmas") with associative
## multiplication.
##
DeclareSynonymAttr( "IsMonoid", IsMagmaWithOne and IsAssociative );
#############################################################################
##
#F Monoid( <gen1>, <gen2> ... )
#F Monoid( <gens> )
#F Monoid( <gens>, <id> )
##
## In the first form, `Monoid' returns the monoid generated by the
## arguments <gen1>, <gen2> ...,
## that is, the closure of these elements under multiplication and taking
## the 0-th power.
## In the second form, `Monoid' returns the monoid generated by the
## elements in the homogeneous list <gens>;
## a square matrix as only argument is treated as one generator,
## not as a list of generators.
## In the third form, `Monoid' returns the monoid generated by the
## elements in the homogeneous list <gens>, with identity <id>.
##
## It is *not* checked whether the underlying multiplication is associative,
## use `MagmaWithOne' (see~"MagmaWithOne") and `IsAssociative'
## (see~"IsAssociative")
## if you want to check whether a magma-with-one is in fact a monoid.
##
DeclareGlobalFunction( "Monoid" );
#############################################################################
##
#F Submonoid( <M>, <gens> ) . . . . . . submonoid of <M> generated by <gens>
#F SubmonoidNC( <M>, <gens> )
##
## are just synonyms of `SubmagmaWithOne' and `SubmagmaWithOneNC',
## respectively (see~"SubmagmaWithOne").
##
DeclareSynonym( "Submonoid", SubmagmaWithOne );
DeclareSynonym( "SubmonoidNC", SubmagmaWithOneNC );
#############################################################################
##
#O MonoidByGenerators( <gens> ) . . . . . . . . . monoid generated by <gens>
#O MonoidByGenerators( <gens>, <one> )
##
## is the underlying operation of `Monoid' (see~"Monoid").
##
DeclareOperation( "MonoidByGenerators", [ IsCollection ] );
#############################################################################
##
#A AsMonoid( <C> ) . . . . . . . . . . . . collection <C> regarded as monoid
##
## If <C> is a collection whose elements form a monoid
## (see~"IsMonoid") then `AsMonoid' returns this monoid.
## Otherwise `fail' is returned.
##
DeclareAttribute( "AsMonoid", IsCollection );
#############################################################################
##
#O AsSubmonoid( <D>, <C> )
##
## Let <D> be a domain and <C> a collection.
## If <C> is a subset of <D> that forms a monoid then `AsSubmonoid'
## returns this monoid, with parent <D>.
## Otherwise `fail' is returned.
##
DeclareOperation( "AsSubmonoid", [ IsDomain, IsCollection ] );
#############################################################################
##
#A GeneratorsOfMonoid( <M> ) . . . . . . . monoid generators of monoid <M>
##
## Monoid generators of a monoid <M> are the same as
## magma-with-one generators (see~"GeneratorsOfMagmaWithOne").
##
DeclareSynonymAttr( "GeneratorsOfMonoid", GeneratorsOfMagmaWithOne );
#############################################################################
##
#A TrivialSubmonoid( <M> ) . . . . . . . . . trivial submonoid of monoid <M>
##
## is just a synonym for `TrivialSubmagmaWithOne'
## (see~"TrivialSubmagmaWithOne").
##
DeclareSynonymAttr( "TrivialSubmonoid", TrivialSubmagmaWithOne );
#############################################################################
##
#F FreeMonoid( [<wfilt>,]<rank> )
#F FreeMonoid( [<wfilt>,]<rank>, <name> )
#F FreeMonoid( [<wfilt>,]<name1>, <name2>, ... )
#F FreeMonoid( [<wfilt>,]<names> )
#F FreeMonoid( [<wfilt>,]infinity, <name>, <init> )
##
## Called in the first form, `FreeMonoid' returns a free monoid on
## <rank> generators.
## Called in the second form, `FreeMonoid' returns a free monoid on
## <rank> generators, printed as `<name>1', `<name>2' etc.,
## that is, each name is the concatenation of the string <name> and an
## integer from `1' to <range>.
## Called in the third form, `FreeMonoid' returns a free monoid on
## as many generators as arguments, printed as <name1>, <name2> etc.
## Called in the fourth form, `FreeMonoid' returns a free monoid on
## as many generators as the length of the list <names>, the $i$-th
## generator being printed as `<names>[$i$]'.
## Called in the fifth form, `FreeMonoid' returns a free monoid on
## infinitely many generators, where the first generators are printed
## by the names in the list <init>, and the other generators by <name>
## and an appended number.
##
## If the extra argument <wfilt> is given, it must be either
## `IsSyllableWordsFamily' or `IsLetterWordsFamily' or
## `IsWLetterWordsFamily' or `IsBLetterWordsFamily'. The filter then
## specifies the representation used for the elements of the free group
## (see~"Representations for Associative Words"). If no such filter is
## given, a letter representation is used.
##
DeclareGlobalFunction( "FreeMonoid" );
#############################################################################
##
#E
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