[ref] 33 Magmas

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# 33 Magmas

### Sections

1. Magma Categories
2. Magma Generation
3. Magmas Defined by Multiplication Tables
4. Attributes and Properties for Magmas

This chapter deals with domains (see Domains and their Elements) that are closed under multiplication `*`. Following Bourbaki70, we call them magmas in GAP. Together with the domains closed under addition `+`, (see Additive Magmas), they are the basic algebraic structures; every semigroup (see Semigroups), monoid (see Monoids), group (see Groups), ring (see Rings), or field (see Fields and Division Rings) is a magma. In the cases of a magma-with-one or magma-with-inverses, additional multiplicative structure is present, see Magma Categories. For functions to create free magmas, see Free Magmas.

## 33.1 Magma Categories

- `IsMagma(` obj`) C`

  A magma in GAP is a domain M with (not necessarily associative) multiplication `*`: M ×M ® M.

- `IsMagmaWithOne(` obj`) C`

  A magma-with-one in GAP is a magma M with an operation `^0` (or `One`) 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`Î` M`` (see MultiplicativeNeutralElement). This element e can be computed with the operation `One` (see One) as `One(` M`)`, and e is also equal to `One(` elm`)` and to ``elm`^0` for each element elm in M.

  Note that a magma may contain a multiplicatively neutral element but not an identity (see One), and a magma containing an identity may not lie in the category `IsMagmaWithOne` (see Domain Categories).

- `IsMagmaWithInversesIfNonzero(` obj`) C`

  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), 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 Additive Magmas), and the set Z, if it is nonempty, contains exactly the zero element (see Zero) of M.

- `IsMagmaWithInverses(` obj`) C`

  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).

  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.

  ## 33.2 Magma Generation

- `Magma(` gens`) F`
- `Magma(` Fam`,` gens`) F`

  returns the magma M that is generated by the elements in the list gens, that is, the closure of gens under multiplication `*`. The family Fam of M can be entered as first argument; this is obligatory if gens is empty (and hence also M is empty).

- `MagmaWithOne(` gens`) F`
- `MagmaWithOne(` Fam`,` gens`) F`

  returns the magma-with-one M that is generated by the elements in the list gens, that is, the closure of gens under multiplication `*` and `One`. The family Fam of M can be entered as first argument; this is obligatory if gens is empty (and hence M is trivial).

- `MagmaWithInverses(` gens`) F`
- `MagmaWithInverses(` Fam`,` gens`) F`

  returns the magma-with-inverses M that is generated by the elements in the list gens, that is, the closure of gens under multiplication `*`, `One`, and `Inverse`. The family Fam of M can be entered as first argument; this is obligatory if gens is empty (and hence M is trivial).

  The underlying operations for which methods can be installed are the following.

- `MagmaByGenerators(` gens`) O`
- `MagmaByGenerators(` Fam`,` gens`) O`

- `MagmaWithOneByGenerators(` gens`) O`
- `MagmaWithOneByGenerators(` Fam`,` generators`) O`

- `MagmaWithInversesByGenerators(` generators`) O`
- `MagmaWithInversesByGenerators(` Fam`,` generators`) O`

  Substructures of a magma can be formed as follows.

- `Submagma(` D`,` gens`) F`
- `SubmagmaNC(` D`,` gens`) F`

  `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.

- `SubmagmaWithOne(` D`,` gens`) F`
- `SubmagmaWithOneNC(` D`,` gens`) F`

  `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.

- `SubmagmaWithInverses(` D`,` gens`) F`
- `SubmagmaWithInversesNC(` D`,` gens`) F`

  `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.

  The following functions can be used to regard a collection as a magma.

- `AsMagma(` C`) A`

  For a collection C whose elements form a magma, `AsMagma` returns this magma. Otherwise `fail` is returned.

- `AsSubmagma(` D`,` C`) O`

  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.

  The following function creates a new magma which is the original magma with a zero adjoined.

- `InjectionZeroMagma(` M`) A`

  The canonical homomorphism i from the magma M into the magma formed from M with a single new element which is a multiplicative zero for the resulting magma.

  The elements of the new magma form a family of elements in the category IsMultiplicativeElementWithZero, and the new magma is obtained as Range(i).

  ## 33.3 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.

- `MagmaByMultiplicationTable(` A`) F`

  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 m1, m2, ¼, mn, and mi * mj = mA[i][j].

  The ordering of elements is defined by m1 < m2 < ¼ < mn, so mi can be accessed as `MagmaElement(` M`,` i`)`, see MagmaElement.

- `MagmaWithOneByMultiplicationTable(` A`) F`

  The only differences between `MagmaByMultiplicationTable` and `MagmaWithOneByMultiplicationTable` are that the latter returns a magma-with-one (see MagmaWithOne) if the magma described by the matrix A has an identity, and returns `fail` if not.

- `MagmaWithInversesByMultiplicationTable(` A`) F`

  `MagmaByMultiplicationTable` and `MagmaWithInversesByMultiplicationTable` differ only in that the latter returns magma-with-inverses (see MagmaWithInverses) if each element in the magma described by the matrix A has an inverse, and returns `fail` if not.

- `MagmaElement(` M`,` i`) F`

  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.

- `MultiplicationTable(` elms`) F`

  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`.

      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

  ## 33.4 Attributes and Properties for Magmas

- `GeneratorsOfMagma(` M`) A`

  is a list gens of elements of the magma M that generates M as a magma, that is, the closure of gens under multiplication is M.

- `GeneratorsOfMagmaWithOne(` M`) A`

  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 and `One` (see One) is M.

- `GeneratorsOfMagmaWithInverses(` M`) A`

  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 and taking inverses (see Inverse) is M.

- `Centralizer(` M`,` elm`) O`
- `Centralizer(` M`,` S`) O`
- `Centralizer(` class`) O`

  For an element elm of the magma M this operation returns the centralizer of elm. This is the domain of those elements ``m`Î` 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` (see ConjugacyClass), `Centralizer` returns the centralizer of `Representative(`class`)` (which is a slight abuse of the notation).

      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(())

- `Centre(` M`) A`
- `Center(` M`) A`

  `Centre` returns the centre of the magma M, i.e., the domain of those elements ``m`Î` M`` that commute with all elements of M. `Center` is just a synonym for `Centre`.

  We have `Centre(` M`) = Centralizer(` M`,` M`)`, see Centralizer.

  The centre of a magma is always commutative (see IsCommutative). (When one installs a new method for `Centre`, one should set the `IsCommutative` value of the result to `true`, in order to make this information available.)

- `Idempotents(` M`) A`

  The set of elements of M which are their own squares.

- `IsAssociative(` M`) P`

  A magma M is associative if for all elements ``a`,` b`,` c`Î` M`` the equality `(`a`*` b`) *` c`=` a`* (`b`*` c`)` holds.

  An associative magma is called a semigroup (see Semigroups), an associative magma-with-one is called a monoid (see Monoids), and an associative magma-with-inverses is called a group (see Groups).

- `IsCentral(` M`,` obj`) O`

  `IsCentral` returns `true` if the object obj, which must either be an element or a magma, commutes with all elements in the magma M.

- `IsCommutative(` M`) P`
- `IsAbelian(` M`) P`

  A magma M is commutative if for all elements ``a`,` b`Î` M`` the equality ``a`*` b`=` b`*` a`` holds. `IsAbelian` is a synonym of `IsCommutative`.

  Note that the commutativity of the addition `+` in an additive structure can be tested with `IsAdditivelyCommutative`, see IsAdditivelyCommutative.

- `MultiplicativeNeutralElement(` M`) A`

  returns the element e in the magma M with the property that ``e`*` m`=` m`=` m`*` e`` holds for all ``m`Î` 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.

- `MultiplicativeZero(` M`) A`

  Returns the multiplicative zero of the magma which is the element z such that for all m in M, ``z`*` m`=` m`*` z`=` z``.

- `IsMultiplicativeZero(` M`,` z`) O`

  returns true iff ``z`*` m`=` m`*` z`=` z`` for all m in M.

- `SquareRoots(` M`,` elm`) O`

  is the proper set of all elements r in the magma M such that ``r`*` r`=` elm`` holds.

- `TrivialSubmagmaWithOne(` M`) A`

  is the magma-with-one that has the identity of the magma-with-one M as only element.

  Note that `IsAssociative` and `IsCommutative` 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 Rings), then the addition `+` is always assumed to be associative and commutative, see Arithmetic Operations for Elements.

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  GAP 4 manual
  May 2002