  
  [1X35 [33X[0;0YMagmas[133X[101X
  
  [33X[0;0YThis   chapter   deals   with   domains   (see [14X31[114X)  that  are  closed  under
  multiplication  [10X*[110X.  Following [Bou70],  we call them [13Xmagmas[113X in [5XGAP[105X. Together
  with  the  domains  closed  under  addition  [10X+[110X  (see [14X55[114X), they are the basic
  algebraic structures; every semigroup, monoid (see [14X51[114X), group (see [14X39[114X), ring
  (see [14X56[114X),  or field (see [14X58[114X) is a magma. In the cases of a [13Xmagma-with-one[113X or
  [13Xmagma-with-inverses[113X,   additional   multiplicative   structure  is  present,
  see [14X35.1[114X. For functions to create free magmas, see [14X36.4[114X.[133X
  
  
  [1X35.1 [33X[0;0YMagma Categories[133X[101X
  
  [1X35.1-1 IsMagma[101X
  
  [29X[2XIsMagma[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YA   [13Xmagma[113X   in  [5XGAP[105X  is  a  domain  [22XM[122X  with  (not  necessarily  associative)
  multiplication [10X*[110X[22X: M × M → M[122X.[133X
  
  [1X35.1-2 IsMagmaWithOne[101X
  
  [29X[2XIsMagmaWithOne[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YA [13Xmagma-with-one[113X in [5XGAP[105X is a magma [22XM[122X with an operation [10X^0[110X (or [2XOne[102X ([14X31.10-2[114X))
  that yields the identity of [22XM[122X.[133X
  
  [33X[0;0YSo  a magma-with-one [22XM[122X does always contain a unique multiplicatively neutral
  element   [22Xe[122X,   i.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      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