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.
‣ IsMagma( obj ) | ( category ) |
A magma in GAP is a domain \(M\) with (not necessarily associative) multiplication *\(: M \times M \rightarrow M\).
‣ 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 \(m \in M\) (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 \(m \in M\).
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).
‣ IsMagmaWithInversesIfNonzero( obj ) | ( category ) |
An object in this GAP category is a magma-with-one \(M\) with an operation ^-1\(: M \setminus Z \rightarrow M \setminus Z\) that maps each element \(m\) of \(M \setminus Z\) 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 \(M\).
‣ IsMagmaWithInverses( obj ) | ( category ) |
A magma-with-inverses in GAP is a magma-with-one \(M\) with an operation ^-1\(: M \rightarrow M\) 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.
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.
‣ 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).
‣ 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).
‣ 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).
‣ MagmaByGenerators( [Fam, ]gens ) | ( operation ) |
An underlying operation for Magma (35.2-1).
‣ MagmaWithOneByGenerators( [Fam, ]gens ) | ( operation ) |
An underlying operation for MagmaWithOne (35.2-2).
‣ MagmaWithInversesByGenerators( [Fam, ]gens ) | ( operation ) |
An underlying operation for MagmaWithInverses (35.2-3).
‣ 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.
‣ 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.
‣ 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.
‣ AsMagma( C ) | ( attribute ) |
For a collection C whose elements form a magma, AsMagma returns this magma. Otherwise fail is returned.
‣ 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.
‣ 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.
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
‣ 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).
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
‣ UnderlyingInjectionZeroMagma( M ) | ( attribute ) |
UnderlyingInjectionZeroMagma returns the embedding used to create the magma with zero adjoined M.
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 )
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 ) | ( 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 \(m_1, m_2, \ldots, m_n\), and \(m_i * m_j = m_k\), with \(k =\) A\([i][j]\).
The ordering of elements is defined by \(m_1 < m_2 < \cdots < m_n\), so \(m_i\) can be accessed as MagmaElement( M, i ), see MagmaElement (35.3-4).
gap> MagmaByMultiplicationTable([[1,2,3],[2,3,1],[1,1,1]]); <magma with 3 generators>
‣ 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.
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
‣ 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.
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
‣ 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.
‣ 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.
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
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.
‣ 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)).
‣ 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)).
‣ 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.
‣ 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 \(\in\) 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).
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 ) | ( attribute ) |
‣ Center( M ) | ( attribute ) |
Centre returns the centre of the magma M, i.e., the domain of those elements m \(\in\) M that commute and associate with all elements of M. That is, the set \(\{ m \in M; \forall a, b \in M: ma = am, (ma)b = m(ab), (am)b = a(mb), (ab)m = a(bm) \}\).
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.)
‣ Idempotents( M ) | ( attribute ) |
The set of elements of M which are their own squares.
‣ IsAssociative( M ) | ( property ) |
A magma M is associative if for all elements \(a, b, c \in\) 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).
‣ 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.
‣ IsCommutative( M ) | ( property ) |
‣ IsAbelian( M ) | ( property ) |
A magma M is commutative if for all elements \(a, b \in\) 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).
‣ MultiplicativeNeutralElement( M ) | ( attribute ) |
returns the element \(e\) in the magma M with the property that \(e\) * \(m = m = m\) * \(e\) holds for all \(m \in\) 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).
‣ 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.
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 ] )
‣ SquareRoots( M, elm ) | ( operation ) |
is the proper set of all elements \(r\) in the magma M such that \(r * r =\) elm holds.
‣ TrivialSubmagmaWithOne( M ) | ( attribute ) |
is the magma-with-one that has the identity of the magma-with-one M as only element.
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