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.
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.
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).
31.3 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
The only differences between MagmaByMultiplicationTable
and
MagmaWithInversesByMultiplicationTable
are that the latter returns a
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
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 ) F
Centralizer(
M,
S ) F
Centralizer(
M,
class ) F
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 M
(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.
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, zm=mz=z.
IsMultiplicativeZero(
M,
z ) O
returns true iff zm = mz = 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.
FreeMagma(
rank ) F
FreeMagma(
rank,
name ) F
FreeMagma(
name1,
name2, ... ) F
FreeMagma(
names ) F
FreeMagma( infinity,
name,
init ) F
Called in the first form, FreeMagma
returns a free magma on
rank generators.
Called in the second form, FreeMagma
returns a free magma on
rank generators, printed as name
1
, name
2
etc.,
that is, each name is the concatenation of the string name and an
integer from 1
to range.
Called in the third form, FreeMagma
returns a free magma on
as many generators as arguments, printed as name1, name2 etc.
Called in the fourth form, FreeMagma
returns a free magma on
as many generators as the length of the list names, the i-th
generator being printed as names
[ i]
.
Called in the fifth form, FreeMagma
returns a free magma on
infinitely many generators, where the first generators are printed
by the names in the list init, and the other generators by name
and an appended number.
FreeMagmaWithOne(
rank ) F
FreeMagmaWithOne(
rank,
name ) F
FreeMagmaWithOne(
name1,
name2, ... ) F
FreeMagmaWithOne(
names ) F
FreeMagmaWithOne( infinity,
name,
init ) F
Called in the first form, FreeMagmaWithOne
returns
a free magma-with-one on rank generators.
Called in the second form, FreeMagmaWithOne
returns
a free magma-with-one on rank generators,
printed as name
1
, name
2
etc.
Called in the third form, FreeMagmaWithOne
returns
a free magma-with-one on as many generators as arguments,
printed as name1, name2 etc.
Called in the fourth form, FreeMagmaWithOne
returns
a free magma-with-one on as many generators as the length of the list
names, the i-th generator being printed as names
[ i]
.
Called in the fifth form, FreeMagmaWithOne
returns
a free magma on infinitely many generators,
where the first generators are printed by the names in the list init,
and the other generators by name and an appended number.
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