35 Magmas
  
  This   chapter   deals   with   domains   (see 31)  that  are  closed  under
  multiplication  *.  Following [Bou70],  we call them magmas in GAP. Together
  with  the  domains  closed  under  addition  +  (see 55), they are the basic
  algebraic structures; every semigroup, monoid (see 51), group (see 39), ring
  (see 56),  or field (see 58) is a magma. In the cases of a magma-with-one or
  magma-with-inverses,   additional   multiplicative   structure  is  present,
  see 35.1. For functions to create free magmas, see 36.4.
  
  
  35.1 Magma Categories
  
  35.1-1 IsMagma
  
  IsMagma( obj )  Category
  
  A   magma   in  GAP  is  a  domain  M  with  (not  necessarily  associative)
  multiplication *: M × M → M.
  
  35.1-2 IsMagmaWithOne
  
  IsMagmaWithOne( obj )  Category
  
  A magma-with-one in GAP is a magma M with an operation ^0 (or One (31.10-2))
  that yields the identity of M.
  
  So  a magma-with-one M does always contain a unique multiplicatively neutral
  element   e,   i.e.,   e   *   m  =  m  =  m  *  e  holds  for  all  m  ∈  M
  (see MultiplicativeNeutralElement (35.4-10)). This element e can be computed
  with  the operation One (31.10-2) as One( M ), and e is also equal to One( m
  ) and to m^0 for each element m ∈ M.
  
  Note  that a magma may contain a multiplicatively neutral element but not an
  identity (see One (31.10-2)), and a magma containing an identity may not lie
  in the category IsMagmaWithOne (see Section 31.6).
  
  35.1-3 IsMagmaWithInversesIfNonzero
  
  IsMagmaWithInversesIfNonzero( obj )  Category
  
  An  object in this GAP category is a magma-with-one M with an operation ^-1:
  M  ∖  Z  →  M  ∖ Z that maps each element m of M ∖ Z to its inverse m^-1 (or
  Inverse(  m  ),  see Inverse (31.10-8)), where Z is either empty or consists
  exactly of one element of M.
  
  This  category  was  introduced mainly to describe division rings, since the
  nonzero  elements  in  a  division  ring  form  a  group;  So an object M in
  IsMagmaWithInversesIfNonzero  will usually have both a multiplicative and an
  additive  structure  (see 55),  and  the  set Z, if it is nonempty, contains
  exactly the zero element (see Zero (31.10-3)) of M.
  
  35.1-4 IsMagmaWithInverses
  
  IsMagmaWithInverses( obj )  Category
  
  A  magma-with-inverses in GAP is a magma-with-one M with an operation ^-1: M
  →  M  that  maps  each  element m of M to its inverse m^-1 (or Inverse( m ),
  see Inverse (31.10-8)).
  
  Note  that  not  every  trivial magma is a magma-with-one, but every trivial
  magma-with-one  is a magma-with-inverses. This holds also if the identity of
  the  magma-with-one  is  a zero element. So a magma-with-inverses-if-nonzero
  can  be  a  magma-with-inverses  if  either  it  contains no zero element or
  consists of a zero element that has itself as zero-th power.
  
  
  35.2 Magma Generation
  
  This  section  describes  functions  that create magmas from generators (see
  Magma  (35.2-1),  MagmaWithOne  (35.2-2),  MagmaWithInverses  (35.2-3)), the
  underlying   operations   for   which   methods   can   be   installed  (see
  MagmaByGenerators      (35.2-4),      MagmaWithOneByGenerators     (35.2-5),
  MagmaWithInversesByGenerators  (35.2-6)),  functions  for  forming submagmas
  (see   Submagma  (35.2-7),  SubmagmaWithOne  (35.2-8),  SubmagmaWithInverses
  (35.2-9)),  and functions that form a magma equal to a given collection (see
  AsMagma (35.2-10), AsSubmagma (35.2-11)).
  
  InjectionZeroMagma (35.2-13) creates a new magma which is the original magma
  with a zero adjoined.
  
  35.2-1 Magma
  
  Magma( [Fam, ]gens )  function
  
  returns the magma M that is generated by the elements in the list gens, that
  is, the closure of gens under multiplication \* (31.12-1). The family Fam of
  M  can be entered as the first argument; this is obligatory if gens is empty
  (and hence also M is empty).
  
  35.2-2 MagmaWithOne
  
  MagmaWithOne( [Fam, ]gens )  function
  
  returns  the  magma-with-one M that is generated by the elements in the list
  gens, that is, the closure of gens under multiplication \* (31.12-1) and One
  (31.10-2).  The  family  Fam  of M can be entered as first argument; this is
  obligatory if gens is empty (and hence M is trivial).
  
  35.2-3 MagmaWithInverses
  
  MagmaWithInverses( [Fam, ]gens )  function
  
  returns  the  magma-with-inverses M that is generated by the elements in the
  list  gens,  that is, the closure of gens under multiplication \* (31.12-1),
  One  (31.10-2), and Inverse (31.10-8). The family Fam of M can be entered as
  first  argument;  this  is  obligatory  if  gens  is  empty  (and hence M is
  trivial).
  
  35.2-4 MagmaByGenerators
  
  MagmaByGenerators( [Fam, ]gens )  operation
  
  An underlying operation for Magma (35.2-1).
  
  35.2-5 MagmaWithOneByGenerators
  
  MagmaWithOneByGenerators( [Fam, ]gens )  operation
  
  An underlying operation for MagmaWithOne (35.2-2).
  
  35.2-6 MagmaWithInversesByGenerators
  
  MagmaWithInversesByGenerators( [Fam, ]gens )  operation
  
  An underlying operation for MagmaWithInverses (35.2-3).
  
  35.2-7 Submagma
  
  Submagma( D, gens )  function
  SubmagmaNC( D, gens )  function
  
  Submagma  returns the magma generated by the elements in the list gens, with
  parent the domain D. SubmagmaNC does the same, except that it is not checked
  whether the elements of gens lie in D.
  
  35.2-8 SubmagmaWithOne
  
  SubmagmaWithOne( D, gens )  function
  SubmagmaWithOneNC( D, gens )  function
  
  SubmagmaWithOne  returns the magma-with-one generated by the elements in the
  list gens, with parent the domain D. SubmagmaWithOneNC does the same, except
  that it is not checked whether the elements of gens lie in D.
  
  35.2-9 SubmagmaWithInverses
  
  SubmagmaWithInverses( D, gens )  function
  SubmagmaWithInversesNC( D, gens )  function
  
  SubmagmaWithInverses   returns  the  magma-with-inverses  generated  by  the
  elements  in the list gens, with parent the domain D. SubmagmaWithInversesNC
  does  the  same,  except that it is not checked whether the elements of gens
  lie in D.
  
  35.2-10 AsMagma
  
  AsMagma( C )  attribute
  
  For  a collection C whose elements form a magma, AsMagma returns this magma.
  Otherwise fail is returned.
  
  35.2-11 AsSubmagma
  
  AsSubmagma( D, C )  operation
  
  Let  D  be  a  domain and C a collection. If C is a subset of D that forms a
  magma  then  AsSubmagma returns this magma, with parent D. Otherwise fail is
  returned.
  
  35.2-12 IsMagmaWithZeroAdjoined
  
  IsMagmaWithZeroAdjoined( M )  Category
  Returns:  true or false.
  
  IsMagmaWithZeroAdjoined  returns  true  if  the  magma  M  was created using
  InjectionZeroMagma  (35.2-13) or MagmaWithZeroAdjoined (35.2-13) and returns
  false if it was not.
  
    Example  
    gap> S:=Semigroup(Transformation([1,1,1]), Transformation([1,3,2]));;
    gap> IsMagmaWithZeroAdjoined(S);
    false
    gap> M:=MagmaWithZeroAdjoined(S);
    <<transformation semigroup of degree 3 with 2 generators>
      with 0 adjoined>
    gap> IsMagmaWithZeroAdjoined(M);
    true
  
  
  35.2-13 InjectionZeroMagma
  
  InjectionZeroMagma( M )  attribute
  MagmaWithZeroAdjoined( M )  attribute
  
  InjectionZeroMagma  returns  an  embedding from the magma M into a new magma
  formed  from M by adjoining a single new element which is the multiplicative
  zero  of the resulting magma. The elements of the new magma form a family of
  elements  in the category IsMultiplicativeElementWithZero (31.14-12) and the
  magma itself satisfies IsMagmaWithZeroAdjoined (35.2-12).
  
  MagmaWithZeroAdjoined is just shorthand for Range(InjectionZeroMagma(M))).
  
  If  N is a magma with zero adjoined, then the embedding used to create N can
  be recovered using UnderlyingInjectionZeroMagma (35.2-14).
  
    Example  
    gap> S:=Monoid(Transformation( [ 7, 7, 5, 3, 1, 3, 7 ] ),
    > Transformation( [ 5, 1, 4, 1, 4, 4, 7 ] ));;
    gap> MultiplicativeZero(S);
    Transformation( [ 7, 7, 7, 7, 7, 7, 7 ] )
    gap> T:=MagmaWithZeroAdjoined(S);
    <<transformation monoid of degree 7 with 2 generators>
      with 0 adjoined>
    gap> map:=UnderlyingInjectionZeroMagma(T);;
    gap> x:=Transformation( [ 7, 7, 7, 3, 7, 3, 7 ] );;
    gap> x^map;
    <monoid with 0 adjoined elt: Transformation( [ 7, 7, 7, 3, 7, 3, 7 ]
     )>
    gap> PreImage(map, x^map)=x;
    true
  
  
  35.2-14 UnderlyingInjectionZeroMagma
  
  UnderlyingInjectionZeroMagma( M )  attribute
  
  UnderlyingInjectionZeroMagma  returns the embedding used to create the magma
  with zero adjoined M.
  
    Example  
    gap> S:=Monoid(Transformation( [ 8, 7, 5, 3, 1, 3, 8, 8 ] ),
    > Transformation( [ 5, 1, 4, 1, 4, 4, 7, 8 ] ));;
    gap> MultiplicativeZero(S);
    Transformation( [ 8, 8, 8, 8, 8, 8, 8, 8 ] )
    gap> T:=MagmaWithZeroAdjoined(S);
    <<transformation monoid of degree 8 with 2 generators>
      with 0 adjoined>
    gap> UnderlyingInjectionZeroMagma(T);
    MappingByFunction( <transformation monoid of degree 8 with 2 
     generators>, <<transformation monoid of degree 8 with 2 generators>
      with 0 adjoined>, function( elt ) ... end, function( x ) ... end )
  
  
  
  35.3 Magmas Defined by Multiplication Tables
  
  The most elementary (but of course usually not recommended) way to implement
  a magma with only few elements is via a multiplication table.
  
  35.3-1 MagmaByMultiplicationTable
  
  MagmaByMultiplicationTable( A )  function
  
  For  a  square  matrix  A  with n rows such that all entries of A are in the
  range  [  1  ..  n  ],  MagmaByMultiplicationTable  returns  a  magma M with
  multiplication * defined by A. That is, M consists of the elements m_1, m_2,
  ..., m_n, and m_i * m_j = m_k, with k = A[i][j].
  
  The  ordering  of  elements is defined by m_1 < m_2 < ⋯ < m_n, so m_i can be
  accessed as MagmaElement( M, i ), see MagmaElement (35.3-4).
  
    Example  
    gap> MagmaByMultiplicationTable([[1,2,3],[2,3,1],[1,1,1]]);
    <magma with 3 generators>
  
  
  35.3-2 MagmaWithOneByMultiplicationTable
  
  MagmaWithOneByMultiplicationTable( A )  function
  
  The   only   differences  between  MagmaByMultiplicationTable  (35.3-1)  and
  MagmaWithOneByMultiplicationTable    are   that   the   latter   returns   a
  magma-with-one  (see MagmaWithOne  (35.2-2))  if  the magma described by the
  matrix A has an identity, and returns fail if not.
  
    Example  
    gap> MagmaWithOneByMultiplicationTable([[1,2,3],[2,3,1],[3,1,1]]);
    <magma-with-one with 3 generators>
    gap> MagmaWithOneByMultiplicationTable([[1,2,3],[2,3,1],[1,1,1]]);
    fail
  
  
  35.3-3 MagmaWithInversesByMultiplicationTable
  
  MagmaWithInversesByMultiplicationTable( A )  function
  
  MagmaByMultiplicationTable                    (35.3-1)                   and
  MagmaWithInversesByMultiplicationTable   differ  only  in  that  the  latter
  returns magma-with-inverses (see MagmaWithInverses (35.2-3)) if each element
  in  the  magma described by the matrix A has an inverse, and returns fail if
  not.
  
    Example  
    gap> MagmaWithInversesByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);
    <magma-with-inverses with 3 generators>
    gap> MagmaWithInversesByMultiplicationTable([[1,2,3],[2,3,1],[3,2,1]]);
    fail
  
  
  35.3-4 MagmaElement
  
  MagmaElement( M, i )  function
  
  For  a  magma  M  and  a  positive  integer i, MagmaElement returns the i-th
  element of M, w.r.t. the ordering <. If M has less than i elements then fail
  is returned.
  
  
  35.3-5 MultiplicationTable
  
  MultiplicationTable( elms )  attribute
  MultiplicationTable( M )  attribute
  
  For a list elms of elements that form a magma M, MultiplicationTable returns
  a  square  matrix  A of positive integers such that A[i][j] = k holds if and
  only  if elms[i] * elms[j] = elms[k]. This matrix can be used to construct a
  magma isomorphic to M, using MagmaByMultiplicationTable (35.3-1).
  
  For   a  magma  M,  MultiplicationTable  returns  the  multiplication  table
  w.r.t. the sorted list of elements of M.
  
    Example  
    gap> l:= [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];;
    gap> a:= MultiplicationTable( l );
    [ [ 1, 2, 3, 4 ], [ 2, 1, 4, 3 ], [ 3, 4, 1, 2 ], [ 4, 3, 2, 1 ] ]
    gap> m:= MagmaByMultiplicationTable( a );
    <magma with 4 generators>
    gap> One( m );
    m1
    gap> elm:= MagmaElement( m, 2 );  One( elm );  elm^2;
    m2
    m1
    m1
    gap> Inverse( elm );
    m2
    gap> AsGroup( m );
    <group of size 4 with 2 generators>
    gap> a:= [ [ 1, 2 ], [ 2, 2 ] ];
    [ [ 1, 2 ], [ 2, 2 ] ]
    gap> m:= MagmaByMultiplicationTable( a );
    <magma with 2 generators>
    gap> One( m );  Inverse( MagmaElement( m, 2 ) );
    m1
    fail
  
  
  
  35.4 Attributes and Properties for Magmas
  
  Note  that IsAssociative (35.4-7) and IsCommutative (35.4-9) always refer to
  the multiplication of a domain. If a magma M has also an additive structure,
  e.g.,  if  M is a ring (see 56), then the addition + is always assumed to be
  associative and commutative, see 31.12.
  
  35.4-1 GeneratorsOfMagma
  
  GeneratorsOfMagma( M )  attribute
  
  is  a list gens of elements of the magma M that generates M as a magma, that
  is, the closure of gens under multiplication \* (31.12-1) is M.
  
  For  a  free magma, each generator can also be accessed using the . operator
  (see GeneratorsOfDomain (31.9-2)).
  
  35.4-2 GeneratorsOfMagmaWithOne
  
  GeneratorsOfMagmaWithOne( M )  attribute
  
  is  a  list  gens  of elements of the magma-with-one M that generates M as a
  magma-with-one,  that  is,  the  closure  of  gens  under  multiplication \*
  (31.12-1) and One (31.10-2) is M.
  
  For  a  free magma with one, each generator can also be accessed using the .
  operator (see GeneratorsOfDomain (31.9-2)).
  
  35.4-3 GeneratorsOfMagmaWithInverses
  
  GeneratorsOfMagmaWithInverses( M )  attribute
  
  is  a list gens of elements of the magma-with-inverses M that generates M as
  a  magma-with-inverses, that is, the closure of gens under multiplication \*
  (31.12-1) and taking inverses (see Inverse (31.10-8)) is M.
  
  
  35.4-4 Centralizer
  
  Centralizer( M, elm )  operation
  Centralizer( M, S )  operation
  Centralizer( class )  attribute
  
  For  an element elm of the magma M this operation returns the centralizer of
  elm. This is the domain of those elements m ∈ M that commute with elm.
  
  For  a  submagma S it returns the domain of those elements that commute with
  all elements s of S.
  
  If  class is a class of objects of a magma (this magma then is stored as the
  ActingDomain   of   class)   such  as  given  by  ConjugacyClass  (39.10-1),
  Centralizer  returns  the  centralizer  of Representative(class) (which is a
  slight abuse of the notation).
  
    Example  
    gap> g:=Group((1,2,3,4),(1,2));;
    gap> Centralizer(g,(1,2,3));
    Group([ (1,2,3) ])
    gap> Centralizer(g,Subgroup(g,[(1,2,3)]));
    Group([ (1,2,3) ])
    gap> Centralizer(g,Subgroup(g,[(1,2,3),(1,2)]));
    Group(())
  
  
  35.4-5 Centre
  
  Centre( M )  attribute
  Center( M )  attribute
  
  Centre returns the centre of the magma M, i.e., the domain of those elements
  m  ∈ M that commute and associate with all elements of M. That is, the set {
  m ∈ M; ∀ a, b ∈ M: ma = am, (ma)b = m(ab), (am)b = a(mb), (ab)m = a(bm) }.
  
  Center is just a synonym for Centre.
  
  For  associative  magmas  we  have  that  Centre( M ) = Centralizer( M, M ),
  see Centralizer (35.4-4).
  
  The  centre  of  a magma is always commutative (see IsCommutative (35.4-9)).
  (When one installs a new method for Centre, one should set the IsCommutative
  (35.4-9)  value  of  the  result  to true, in order to make this information
  available.)
  
  35.4-6 Idempotents
  
  Idempotents( M )  attribute
  
  The set of elements of M which are their own squares.
  
  35.4-7 IsAssociative
  
  IsAssociative( M )  property
  
  A  collection  M  of  elements  that  can  be multiplied via \* (31.12-1) is
  associative  if  for all elements a, b, c ∈ M the equality (a * b) * c = a *
  (b * c) holds.
  
  An  associative  magma  is  called  a  semigroup  (see 51),  an  associative
  magma-with-one   is   called   a   monoid   (see 51),   and  an  associative
  magma-with-inverses is called a group (see 39).
  
  35.4-8 IsCentral
  
  IsCentral( M, obj )  operation
  
  IsCentral returns true if the object obj, which must either be an element or
  a magma, commutes with all elements in the magma M.
  
  35.4-9 IsCommutative
  
  IsCommutative( M )  property
  IsAbelian( M )  property
  
  A  collection  M  of  elements  that  can  be multiplied via \* (31.12-1) is
  commutative  if  for all elements a, b ∈ M the equality a * b = b * a holds.
  IsAbelian is a synonym of IsCommutative.
  
  Note  that  the  commutativity  of  the addition \+ (31.12-1) in an additive
  structure can be tested with IsAdditivelyCommutative (55.3-1).
  
  35.4-10 MultiplicativeNeutralElement
  
  MultiplicativeNeutralElement( M )  attribute
  
  returns  the element e in the magma M with the property that e * m = m = m *
  e  holds  for  all  m  ∈  M,  if  such  an element exists. Otherwise fail is
  returned.
  
  A  magma  that  is  not  a  magma-with-one can have a multiplicative neutral
  element  e;  in  this  case,  e  cannot  be  obtained  as  One( M ), see One
  (31.10-2).
  
  35.4-11 MultiplicativeZero
  
  MultiplicativeZero( M )  attribute
  IsMultiplicativeZero( M, z )  operation
  
  MultiplicativeZero  returns  the multiplicative zero of the magma M which is
  the element z in M such that z * m = m * z = z for all m in M.
  
  IsMultiplicativeZero returns true if the element z of the magma M equals the
  multiplicative zero of M.
  
    Example  
    gap> S:=Semigroup( Transformation( [ 1, 1, 1 ] ), 
    > Transformation( [ 2, 3, 1 ] ) );
    <transformation semigroup of degree 3 with 2 generators>
    gap> MultiplicativeZero(S);
    fail
    gap> S:=Semigroup( Transformation( [ 1, 1, 1 ] ), 
    > Transformation( [ 1, 3, 2 ] ) );
    <transformation semigroup of degree 3 with 2 generators>
    gap> MultiplicativeZero(S);
    Transformation( [ 1, 1, 1 ] )
  
  
  35.4-12 SquareRoots
  
  SquareRoots( M, elm )  operation
  
  is  the  proper  set  of all elements r in the magma M such that r * r = elm
  holds.
  
  35.4-13 TrivialSubmagmaWithOne
  
  TrivialSubmagmaWithOne( M )  attribute
  
  is  the magma-with-one that has the identity of the magma-with-one M as only
  element.