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  55 Additive Magmas
  
  This  chapter deals with domains that are closed under addition +, which are
  called  near-additive  magmas in GAP. Together with the domains closed under
  multiplication  * (see 35), they are the basic algebraic structures. In many
  cases,  the  addition is commutative (see IsAdditivelyCommutative (55.3-1)),
  the  domain  is called an additive magma then. Every module (see 57), vector
  space  (see 61),  ring  (see 56), or field (see 58) is an additive magma. In
  the   cases   of  all  (near-)additive  magma-with-zero  or  (near-)additive
  magma-with-inverses, additional additive structure is present (see 55.1).
  
  
  55.1 (Near-)Additive Magma Categories
  
  55.1-1 IsNearAdditiveMagma
  
  IsNearAdditiveMagma( obj )  Category
  
  A  near-additive  magma  in  GAP  is  a domain A with an associative but not
  necessarily commutative addition +: A × A → A.
  
  55.1-2 IsNearAdditiveMagmaWithZero
  
  IsNearAdditiveMagmaWithZero( obj )  Category
  
  A  near-additive  magma-with-zero  in GAP is a near-additive magma A with an
  operation 0* (or Zero (31.10-3)) that yields the zero element of A.
  
  So a near-additive magma-with-zero A does always contain a unique additively
  neutral  element  z,  i.e.,  z  +  a  =  a  =  a  +  z  holds  for all a ∈ A
  (see AdditiveNeutralElement  (55.3-5)).  This zero element z can be computed
  with  the operation Zero (31.10-3), by applying this function to A or to any
  element a in A. The zero element can be computed also as 0 * a, for any a in
  A.
  
  Note  that  it  may happen that a near-additive magma containing a zero does
  not lie in the category IsNearAdditiveMagmaWithZero (see 31.6).
  
  55.1-3 IsNearAdditiveGroup
  
  IsNearAdditiveGroup( obj )  Category
  IsNearAdditiveMagmaWithInverses( obj )  Category
  
  A  near-additive  group  in GAP is a near-additive magma-with-zero A with an
  operation  -1*:  A → A that maps each element a of A to its additive inverse
  -1*a (or AdditiveInverse( a ), see AdditiveInverse (31.10-9)).
  
  The  addition  + of A is assumed to be associative, so a near-additive group
  is     not     more     than     a     near-additive    magma-with-inverses.
  IsNearAdditiveMagmaWithInverses  is  just a synonym for IsNearAdditiveGroup,
  and  can  be  used  alternatively in all function names involving the string
  "NearAdditiveGroup".
  
  Note   that  not  every  trivial  near-additive  magma  is  a  near-additive
  magma-with-zero,  but  every  trivial  near-additive  magma-with-zero  is  a
  near-additive group.
  
  55.1-4 IsAdditiveMagma
  
  IsAdditiveMagma( obj )  Category
  
  An  additive  magma in GAP is a domain A with an associative and commutative
  addition   +:   A   ×   A   →   A,   see IsNearAdditiveMagma   (55.1-1)  and
  IsAdditivelyCommutative (55.3-1).
  
  55.1-5 IsAdditiveMagmaWithZero
  
  IsAdditiveMagmaWithZero( obj )  Category
  
  An   additive   magma-with-zero   in   GAP  is  an  additive  magma  A  (see
  IsAdditiveMagma  (55.1-4)  with  an  operation  0*  (or Zero (31.10-3)) that
  yields the zero of A.
  
  So  an  additive  magma-with-zero  A does always contain a unique additively
  neutral  element  z,  i.e.,  z  +  a  =  a  =  a  +  z  holds  for all a ∈ A
  (see AdditiveNeutralElement  (55.3-5)).  This element z can be computed with
  the  operation Zero (31.10-3) as Zero( A ), and z is also equal to Zero( a )
  and to 0*a for each element a in A.
  
  Note  that  it  may happen that an additive magma containing a zero does not
  lie in the category IsAdditiveMagmaWithZero (see 31.6).
  
  55.1-6 IsAdditiveGroup
  
  IsAdditiveGroup( obj )  Category
  IsAdditiveMagmaWithInverses( obj )  Category
  
  An  additive group in GAP is an additive magma-with-zero A with an operation
  -1*:  A  →  A that maps each element a of A to its additive inverse -1*a (or
  AdditiveInverse( a ), see AdditiveInverse (31.10-9)).
  
  The  addition  +  of  A  is assumed to be commutative and associative, so an
  additive   group   is   not   more  than  an  additive  magma-with-inverses.
  IsAdditiveMagmaWithInverses  is  just a synonym for IsAdditiveGroup, and can
  be   used   alternatively   in  all  function  names  involving  the  string
  "AdditiveGroup".
  
  Note  that  not every trivial additive magma is an additive magma-with-zero,
  but every trivial additive magma-with-zero is an additive group.
  
  
  55.2 (Near-)Additive Magma Generation
  
  This section describes functions that create additive magmas from generators
  (see   NearAdditiveMagma   (55.2-1),   NearAdditiveMagmaWithZero   (55.2-2),
  NearAdditiveGroup (55.2-3)), the underlying operations for which methods can
  be      installed      (see      NearAdditiveMagmaByGenerators     (55.2-4),
  NearAdditiveMagmaWithZeroByGenerators                              (55.2-5),
  NearAdditiveGroupByGenerators  (55.2-6))  and functions for forming additive
  submagmas  (see  SubnearAdditiveMagma (55.2-7), SubnearAdditiveMagmaWithZero
  (55.2-8), SubnearAdditiveGroup (55.2-9)).
  
  55.2-1 NearAdditiveMagma
  
  NearAdditiveMagma( [Fam, ]gens )  function
  
  returns the (near-)additive magma A that is generated by the elements in the
  list  gens, that is, the closure of gens under addition +. The family Fam of
  A can be entered as first argument; this is obligatory if gens is empty (and
  hence also A is empty).
  
  55.2-2 NearAdditiveMagmaWithZero
  
  NearAdditiveMagmaWithZero( [Fam, ]gens )  function
  
  returns  the  (near-)additive  magma-with-zero  A  that  is generated by the
  elements in the list gens, that is, the closure of gens under addition + and
  Zero  (31.10-3).  The family Fam of A can be entered as first argument; this
  is obligatory if gens is empty (and hence A is trivial).
  
  55.2-3 NearAdditiveGroup
  
  NearAdditiveGroup( [Fam, ]gens )  function
  
  returns the (near-)additive group A that is generated by the elements in the
  list  gens,  that  is, the closure of gens under addition +, Zero (31.10-3),
  and  AdditiveInverse  (31.10-9). The family Fam of A can be entered as first
  argument; this is obligatory if gens is empty (and hence A is trivial).
  
  55.2-4 NearAdditiveMagmaByGenerators
  
  NearAdditiveMagmaByGenerators( [Fam, ]gens )  operation
  
  An underlying operation for NearAdditiveMagma (55.2-1).
  
  55.2-5 NearAdditiveMagmaWithZeroByGenerators
  
  NearAdditiveMagmaWithZeroByGenerators( [Fam, ]gens )  operation
  
  An underlying operation for NearAdditiveMagmaWithZero (55.2-2).
  
  55.2-6 NearAdditiveGroupByGenerators
  
  NearAdditiveGroupByGenerators( [Fam, ]gens )  operation
  
  An underlying operation for NearAdditiveGroup (55.2-3).
  
  55.2-7 SubnearAdditiveMagma
  
  SubnearAdditiveMagma( D, gens )  function
  SubadditiveMagma( D, gens )  function
  SubnearAdditiveMagmaNC( D, gens )  function
  SubadditiveMagmaNC( D, gens )  function
  
  SubnearAdditiveMagma  returns  the  near-additive  magma  generated  by  the
  elements  in the list gens, with parent the domain D. SubnearAdditiveMagmaNC
  does  the  same,  except that it does not check whether the elements of gens
  lie in D.
  
  SubadditiveMagma   and   SubadditiveMagmaNC   are  just  synonyms  of  these
  functions.
  
  55.2-8 SubnearAdditiveMagmaWithZero
  
  SubnearAdditiveMagmaWithZero( D, gens )  function
  SubadditiveMagmaWithZero( D, gens )  function
  SubnearAdditiveMagmaWithZeroNC( D, gens )  function
  SubadditiveMagmaWithZeroNC( D, gens )  function
  
  SubnearAdditiveMagmaWithZero   returns   the  near-additive  magma-with-zero
  generated  by  the  elements  in  the  list  gens, with parent the domain D.
  SubnearAdditiveMagmaWithZeroNC  does the same, except that it does not check
  whether the elements of gens lie in D.
  
  SubadditiveMagmaWithZero and SubadditiveMagmaWithZeroNC are just synonyms of
  these functions.
  
  55.2-9 SubnearAdditiveGroup
  
  SubnearAdditiveGroup( D, gens )  function
  SubadditiveGroup( D, gens )  function
  SubnearAdditiveGroupNC( D, gens )  function
  SubadditiveGroupNC( D, gens )  function
  
  SubnearAdditiveGroup  returns  the  near-additive  group  generated  by  the
  elements in the list gens, with parent the domain D. SubadditiveGroupNC does
  the  same, except that it does not check whether the elements of gens lie in
  D.
  
  SubadditiveGroup   and   SubadditiveGroupNC   are  just  synonyms  of  these
  functions.
  
  
  55.3 Attributes and Properties for (Near-)Additive Magmas
  
  55.3-1 IsAdditivelyCommutative
  
  IsAdditivelyCommutative( A )  property
  
  A near-additive magma A in GAP is additively commutative if for all elements
  a, b ∈ A the equality a + b = b + a holds.
  
  Note  that  the  commutativity  of  the multiplication * in a multiplicative
  structure can be tested with IsCommutative (35.4-9).
  
  55.3-2 GeneratorsOfNearAdditiveMagma
  
  GeneratorsOfNearAdditiveMagma( A )  attribute
  GeneratorsOfAdditiveMagma( A )  attribute
  
  is  a  list  of  elements of the near-additive magma A that generates A as a
  near-additive magma, that is, the closure of this list under addition is A.
  
  55.3-3 GeneratorsOfNearAdditiveMagmaWithZero
  
  GeneratorsOfNearAdditiveMagmaWithZero( A )  attribute
  GeneratorsOfAdditiveMagmaWithZero( A )  attribute
  
  is  a list of elements of the near-additive magma-with-zero A that generates
  A  as  a  near-additive  magma-with-zero,  that is, the closure of this list
  under addition and Zero (31.10-3) is A.
  
  55.3-4 GeneratorsOfNearAdditiveGroup
  
  GeneratorsOfNearAdditiveGroup( A )  attribute
  GeneratorsOfAdditiveGroup( A )  attribute
  
  is  a  list  of  elements of the near-additive group A that generates A as a
  near-additive  group,  that  is,  the  closure  of this list under addition,
  taking  the  zero element, and taking additive inverses (see AdditiveInverse
  (31.10-9)) is A.
  
  55.3-5 AdditiveNeutralElement
  
  AdditiveNeutralElement( A )  attribute
  
  returns  the element z in the near-additive magma A with the property that z
  +  a  =  a = a + z holds for all a ∈ A, if such an element exists. Otherwise
  fail is returned.
  
  A  near-additive  magma that is not a near-additive magma-with-zero can have
  an additive neutral element z; in this case, z cannot be obtained as Zero( A
  ) or as 0*a for an element a in A, see Zero (31.10-3).
  
  55.3-6 TrivialSubnearAdditiveMagmaWithZero
  
  TrivialSubnearAdditiveMagmaWithZero( A )  attribute
  
  is  the  additive  magma-with-zero  that  has  the zero of the near-additive
  magma-with-zero A as its only element.
  
  
  55.4 Operations for (Near-)Additive Magmas
  
  
  55.4-1 ClosureNearAdditiveGroup
  
  ClosureNearAdditiveGroup( A, a )  operation
  ClosureNearAdditiveGroup( A, B )  operation
  
  returns  the closure of the near-additive magma A with the element a or with
  the  near-additive  magma  B,  w.r.t. addition, taking the zero element, and
  taking additive inverses.
  
  55.4-2 ShowAdditionTable
  
  ShowAdditionTable( R )  function
  ShowMultiplicationTable( M )  function
  
  For  a  structure  R with an addition given by +, respectively a structure M
  with  a  multiplication  given  by  *,  this  command  displays the addition
  (multiplication) table of the structure in a pretty way.
  
    Example  
    gap> ShowAdditionTable(GF(4));
    +        | 0*Z(2)   Z(2)^0   Z(2^2)   Z(2^2)^2
    ---------+------------------------------------
    0*Z(2)   | 0*Z(2)   Z(2)^0   Z(2^2)   Z(2^2)^2
    Z(2)^0   | Z(2)^0   0*Z(2)   Z(2^2)^2 Z(2^2)  
    Z(2^2)   | Z(2^2)   Z(2^2)^2 0*Z(2)   Z(2)^0  
    Z(2^2)^2 | Z(2^2)^2 Z(2^2)   Z(2)^0   0*Z(2)  
    
    gap> ShowMultiplicationTable(GF(4));             
    *        | 0*Z(2)   Z(2)^0   Z(2^2)   Z(2^2)^2
    ---------+------------------------------------
    0*Z(2)   | 0*Z(2)   0*Z(2)   0*Z(2)   0*Z(2)  
    Z(2)^0   | 0*Z(2)   Z(2)^0   Z(2^2)   Z(2^2)^2
    Z(2^2)   | 0*Z(2)   Z(2^2)   Z(2^2)^2 Z(2)^0  
    Z(2^2)^2 | 0*Z(2)   Z(2^2)^2 Z(2)^0   Z(2^2)