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  52 Monoids
  
  This  chapter  describes functions for monoids. Currently there are only few
  of  them.  More  general functions for magmas and semigroups can be found in
  Chapters 35 and 51.
  
  
  52.1 Functions for Monoids
  
  52.1-1 IsMonoid
  
  IsMonoid( D )  property
  
  A monoid is a magma-with-one (see 35) with associative multiplication.
  
  
  52.1-2 Monoid
  
  Monoid( gen1, gen2, ... )  function
  Monoid( gens[, id] )  function
  
  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  second  form,  the  identity  element  id  may  be  given as the second
  argument.
  
  It  is not checked whether the underlying multiplication is associative, use
  MagmaWithOne  (35.2-2)  and  IsAssociative  (35.4-7)  if  you  want to check
  whether a magma-with-one is in fact a monoid.
  
  52.1-3 Submonoid
  
  Submonoid( M, gens )  function
  SubmonoidNC( M, gens )  function
  
  are   just   synonyms  of  SubmagmaWithOne  (35.2-8)  and  SubmagmaWithOneNC
  (35.2-8), respectively.
  
  52.1-4 MonoidByGenerators
  
  MonoidByGenerators( gens[, one] )  operation
  
  is the underlying operation of Monoid (52.1-2).
  
  52.1-5 AsMonoid
  
  AsMonoid( C )  attribute
  
  If  C  is  a collection whose elements form a monoid (see IsMonoid (52.1-1))
  then AsMonoid returns this monoid. Otherwise fail is returned.
  
  52.1-6 AsSubmonoid
  
  AsSubmonoid( D, C )  operation
  
  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.
  
  52.1-7 GeneratorsOfMonoid
  
  GeneratorsOfMonoid( M )  attribute
  
  Monoid  generators  of  a monoid M are the same as magma-with-one generators
  (see GeneratorsOfMagmaWithOne (35.4-2)).
  
  52.1-8 TrivialSubmonoid
  
  TrivialSubmonoid( M )  attribute
  
  is just a synonym for TrivialSubmagmaWithOne (35.4-14).
  
  
  52.1-9 FreeMonoid
  
  FreeMonoid( [wfilt, ]rank[, name] )  function
  FreeMonoid( [wfilt, ]name1, name2, ... )  function
  FreeMonoid( [wfilt, ]names )  function
  FreeMonoid( [wfilt, ]infinity, name, init )  function
  
  Called  with  a  positive  integer rank, FreeMonoid returns a free monoid on
  rank  generators. If the optional argument name is given then the generators
  are printed as name1, name2 etc., that is, each name is the concatenation of
  the  string name and an integer from 1 to range. The default for name is the
  string "m".
  
  Called  in  the  second  form,  FreeMonoid  returns a free monoid on as many
  generators as arguments, printed as name1, name2 etc.
  
  Called  in  the  third  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  fourth 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    (37.6-6)   or   IsLetterWordsFamily   (37.6-2)   or
  IsWLetterWordsFamily  (37.6-4) or IsBLetterWordsFamily (37.6-4). This filter
  then  specifies  the representation used for the elements of the free monoid
  (see 37.6). If no such filter is given, a letter representation is used.
  
  Also see Chapter 52.
  
  52.1-10 MonoidByMultiplicationTable
  
  MonoidByMultiplicationTable( A )  function
  
  returns  the  monoid  whose multiplication is defined by the square matrix A
  (see MagmaByMultiplicationTable (35.3-1)) if such a monoid exists. Otherwise
  fail is returned.

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