Magmas¶

class
sage.categories.magmas.
Magmas
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
The category of (multiplicative) magmas.
A magma is a set with a binary operation \(*\).
EXAMPLES:
sage: Magmas() Category of magmas sage: Magmas().super_categories() [Category of sets] sage: Magmas().all_super_categories() [Category of magmas, Category of sets, Category of sets with partial maps, Category of objects]
The following axioms are defined by this category:
sage: Magmas().Associative() Category of semigroups sage: Magmas().Unital() Category of unital magmas sage: Magmas().Commutative() Category of commutative magmas sage: Magmas().Unital().Inverse() Category of inverse unital magmas sage: Magmas().Associative() Category of semigroups sage: Magmas().Associative().Unital() Category of monoids sage: Magmas().Associative().Unital().Inverse() Category of groups

class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory

class
ParentMethods
¶ Bases:
object

is_field
(proof=True)¶ Return
True
ifself
is a field.For a magma algebra \(RS\) this is always false unless \(S\) is trivial and the base ring \(R\) is a field.
EXAMPLES:
sage: SymmetricGroup(1).algebra(QQ).is_field() True sage: SymmetricGroup(1).algebra(ZZ).is_field() False sage: SymmetricGroup(2).algebra(QQ).is_field() False


extra_super_categories
()¶ EXAMPLES:
sage: Magmas().Commutative().Algebras(QQ).extra_super_categories() [Category of commutative magmas]
This implements the fact that the algebra of a commutative magma is commutative:
sage: Magmas().Commutative().Algebras(QQ).super_categories() [Category of magma algebras over Rational Field, Category of commutative magmas]
In particular, commutative monoid algebras are commutative algebras:
sage: Monoids().Commutative().Algebras(QQ).is_subcategory(Algebras(QQ).Commutative()) True

class

Associative
¶

class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory

class
ParentMethods
¶ Bases:
object

product
(left, right)¶ EXAMPLES:
sage: C = Magmas().CartesianProducts().example(); C The Cartesian product of (Rational Field, Integer Ring, Integer Ring) sage: x = C.an_element(); x (1/2, 1, 1) sage: x * x (1/4, 1, 1) sage: A = SymmetricGroupAlgebra(QQ, 3) sage: x = cartesian_product([A([1,3,2]), A([2,3,1])]) sage: y = cartesian_product([A([1,3,2]), A([2,3,1])]) sage: cartesian_product([A,A]).product(x,y) B[(0, [1, 2, 3])] + B[(1, [3, 1, 2])] sage: x*y B[(0, [1, 2, 3])] + B[(1, [3, 1, 2])]


example
()¶ Return an example of Cartesian product of magmas.
EXAMPLES:
sage: C = Magmas().CartesianProducts().example(); C The Cartesian product of (Rational Field, Integer Ring, Integer Ring) sage: C.category() Join of Category of Cartesian products of commutative rings and Category of Cartesian products of metric spaces sage: sorted(C.category().axioms()) ['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital', 'Associative', 'Commutative', 'Distributive', 'Unital'] sage: TestSuite(C).run()

extra_super_categories
()¶ This implements the fact that a subquotient (and therefore a quotient or subobject) of a finite set is finite.
EXAMPLES:
sage: Semigroups().CartesianProducts().extra_super_categories() [Category of semigroups] sage: Semigroups().CartesianProducts().super_categories() [Category of semigroups, Category of Cartesian products of magmas]

class

class
Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton

class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory

extra_super_categories
()¶ EXAMPLES:
sage: Magmas().Commutative().Algebras(QQ).extra_super_categories() [Category of commutative magmas]
This implements the fact that the algebra of a commutative magma is commutative:
sage: Magmas().Commutative().Algebras(QQ).super_categories() [Category of magma algebras over Rational Field, Category of commutative magmas]
In particular, commutative monoid algebras are commutative algebras:
sage: Monoids().Commutative().Algebras(QQ).is_subcategory(Algebras(QQ).Commutative()) True


class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory

extra_super_categories
()¶ Implement the fact that a Cartesian product of commutative additive magmas is still an commutative additive magmas.
EXAMPLES:
sage: C = Magmas().Commutative().CartesianProducts() sage: C.extra_super_categories() [Category of commutative magmas] sage: C.axioms() frozenset({'Commutative'})


class

class
ElementMethods
¶ Bases:
object

is_idempotent
()¶ Test whether
self
is idempotent.EXAMPLES:
sage: S = Semigroups().example("free"); S An example of a semigroup: the free semigroup generated by ('a', 'b', 'c', 'd') sage: a = S('a') sage: a^2 'aa' sage: a.is_idempotent() False
sage: L = Semigroups().example("leftzero"); L An example of a semigroup: the left zero semigroup sage: x = L('x') sage: x^2 'x' sage: x.is_idempotent() True


FinitelyGeneratedAsMagma
¶ alias of
sage.categories.finitely_generated_magmas.FinitelyGeneratedMagmas

class
JTrivial
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
ParentMethods
¶ Bases:
object

multiplication_table
(names='letters', elements=None)¶ Returns a table describing the multiplication operation.
Note
The order of the elements in the row and column headings is equal to the order given by the table’s
list()
method. The association can also be retrieved with thedict()
method.INPUT:
names
 the type of names used'letters'
 lowercase ASCII letters are used for a base 26 representation of the elements’ positions in the list given bycolumn_keys()
, padded to a common width with leading ‘a’s.'digits'
 base 10 representation of the elements’ positions in the list given bycolumn_keys()
, padded to a common width with leading zeros.'elements'
 the string representations of the elements themselves.a list  a list of strings, where the length of the list equals the number of elements.
elements
 default =None
. A list of elements of the magma, in forms that can be coerced into the structure, eg. their string representations. This may be used to impose an alternate ordering on the elements, perhaps when this is used in the context of a particular structure. The default is to use whatever ordering theS.list
method returns. Or theelements
can be a subset which is closed under the operation. In particular, this can be used when the base set is infinite.
OUTPUT: The multiplication table as an object of the class
OperationTable
which defines several methods for manipulating and displaying the table. See the documentation there for full details to supplement the documentation here.EXAMPLES:
The default is to represent elements as lowercase ASCII letters.
sage: G = CyclicPermutationGroup(5) sage: G.multiplication_table() * a b c d e + a a b c d e b b c d e a c c d e a b d d e a b c e e a b c d
All that is required is that an algebraic structure has a multiplication defined. A
LeftRegularBand
is an example of a finite semigroup. Thenames
argument allows displaying the elements in different ways.sage: from sage.categories.examples.finite_semigroups import LeftRegularBand sage: L = LeftRegularBand(('a', 'b')) sage: T = L.multiplication_table(names='digits') sage: T.column_keys() ('a', 'ab', 'b', 'ba') sage: T * 0 1 2 3 + 0 0 1 1 1 1 1 1 1 1 2 3 3 2 3 3 3 3 3 3
Specifying the elements in an alternative order can provide more insight into how the operation behaves.
sage: L = LeftRegularBand(('a', 'b', 'c')) sage: elts = sorted(L.list()) sage: L.multiplication_table(elements=elts) * a b c d e f g h i j k l m n o + a a b c d e b b c c c d d e e e b b b c c c b b c c c c c c c c c c c c c c c c c c c c c c c c d d e e d e e e e e e d d e e e e e e e e e e e e e e e e e e e f g g h h h f g h i j i j j i j g g g h h h g g h h h h h h h h h h h h h h h h h h h h h h h h i j j j j j i j j i j i j j i j j j j j j j j j j j j j j j j j k l m m l m n o o n o k l m n o l l m m l m m m m m m l l m m m m m m m m m m m m m m m m m m m n o o o o o n o o n o n o o n o o o o o o o o o o o o o o o o o
The
elements
argument can be used to provide a subset of the elements of the structure. The subset must be closed under the operation. Elements need only be in a form that can be coerced into the set. Thenames
argument can also be used to request that the elements be represented with their usual string representation.sage: L=LeftRegularBand(('a','b','c')) sage: elts=['a', 'c', 'ac', 'ca'] sage: L.multiplication_table(names='elements', elements=elts) * 'a' 'c' 'ac' 'ca' + 'a' 'a' 'ac' 'ac' 'ac' 'c' 'ca' 'c' 'ca' 'ca' 'ac' 'ac' 'ac' 'ac' 'ac' 'ca' 'ca' 'ca' 'ca' 'ca'
The table returned can be manipulated in various ways. See the documentation for
OperationTable
for more comprehensive documentation.sage: G=AlternatingGroup(3) sage: T=G.multiplication_table() sage: T.column_keys() ((), (1,2,3), (1,3,2)) sage: T.translation() {'a': (), 'b': (1,2,3), 'c': (1,3,2)} sage: T.change_names(['x', 'y', 'z']) sage: T.translation() {'x': (), 'y': (1,2,3), 'z': (1,3,2)} sage: T * x y z + x x y z y y z x z z x y

product
(x, y)¶ The binary multiplication of the magma.
INPUT:
x
,y
– elements of this magma
OUTPUT:
an element of the magma (the product of
x
andy
)
EXAMPLES:
sage: S = Semigroups().example("free") sage: x = S('a'); y = S('b') sage: S.product(x, y) 'ab'
A parent in
Magmas()
must either implementproduct()
in the parent class or_mul_
in the element class. By default, the addition method on elementsx._mul_(y)
callsS.product(x,y)
, and reciprocally.As a bonus,
S.product
models the binary function fromS
toS
:sage: bin = S.product sage: bin(x,y) 'ab'
Currently,
S.product
is just a bound method:sage: bin # py2 <bound method FreeSemigroup_with_category.product of An example of a semigroup: the free semigroup generated by ('a', 'b', 'c', 'd')> sage: bin # py3, due to difference in how bound methods are repr'd <bound method FreeSemigroup.product of An example of a semigroup: the free semigroup generated by ('a', 'b', 'c', 'd')>
When Sage will support multivariate morphisms, it will be possible, and in fact recommended, to enrich
S.product
with extra mathematical structure. This will typically be implemented using lazy attributes.:sage: bin # todo: not implemented Generic binary morphism: From: (S x S) To: S

product_from_element_class_mul
(x, y)¶ The binary multiplication of the magma.
INPUT:
x
,y
– elements of this magma
OUTPUT:
an element of the magma (the product of
x
andy
)
EXAMPLES:
sage: S = Semigroups().example("free") sage: x = S('a'); y = S('b') sage: S.product(x, y) 'ab'
A parent in
Magmas()
must either implementproduct()
in the parent class or_mul_
in the element class. By default, the addition method on elementsx._mul_(y)
callsS.product(x,y)
, and reciprocally.As a bonus,
S.product
models the binary function fromS
toS
:sage: bin = S.product sage: bin(x,y) 'ab'
Currently,
S.product
is just a bound method:sage: bin # py2 <bound method FreeSemigroup_with_category.product of An example of a semigroup: the free semigroup generated by ('a', 'b', 'c', 'd')> sage: bin # py3, due to difference in how bound methods are repr'd <bound method FreeSemigroup.product of An example of a semigroup: the free semigroup generated by ('a', 'b', 'c', 'd')>
When Sage will support multivariate morphisms, it will be possible, and in fact recommended, to enrich
S.product
with extra mathematical structure. This will typically be implemented using lazy attributes.:sage: bin # todo: not implemented Generic binary morphism: From: (S x S) To: S


class
Realizations
(category, *args)¶ Bases:
sage.categories.realizations.RealizationsCategory

class
ParentMethods
¶ Bases:
object

product_by_coercion
(left, right)¶ Default implementation of product for realizations.
This method coerces to the realization specified by
self.realization_of().a_realization()
, computes the product in that realization, and then coerces back.EXAMPLES:
sage: Out = Sets().WithRealizations().example().Out(); Out The subset algebra of {1, 2, 3} over Rational Field in the Out basis sage: Out.product <bound method SubsetAlgebra.Out_with_category.product_by_coercion of The subset algebra of {1, 2, 3} over Rational Field in the Out basis> sage: Out.product.__module__ 'sage.categories.magmas' sage: x = Out.an_element() sage: y = Out.an_element() sage: Out.product(x, y) Out[{}] + 4*Out[{1}] + 9*Out[{2}] + Out[{1, 2}]


class

class
SubcategoryMethods
¶ Bases:
object

Associative
()¶ Return the full subcategory of the associative objects of
self
.A (multiplicative)
magma Magmas
\(M\) is associative if, for all \(x,y,z\in M\),\[x * (y * z) = (x * y) * z\]EXAMPLES:
sage: Magmas().Associative() Category of semigroups

Commutative
()¶ Return the full subcategory of the commutative objects of
self
.A (multiplicative)
magma Magmas
\(M\) is commutative if, for all \(x,y\in M\),\[x * y = y * x\]EXAMPLES:
sage: Magmas().Commutative() Category of commutative magmas sage: Monoids().Commutative() Category of commutative monoids

Distributive
()¶ Return the full subcategory of the objects of
self
where \(*\) is distributive on \(+\).INPUT:
self
– a subcategory ofMagmas
andAdditiveMagmas
Given that Sage does not yet know that the category
MagmasAndAdditiveMagmas
is the intersection of the categoriesMagmas
andAdditiveMagmas
, the methodMagmasAndAdditiveMagmas.SubcategoryMethods.Distributive()
is not available, as would be desirable, for this intersection.This method is a workaround. It checks that
self
is a subcategory of bothMagmas
andAdditiveMagmas
and upgrades it to a subcategory ofMagmasAndAdditiveMagmas
before applying the axiom. It complains overwise, since theDistributive
axiom does not make sense for a plain magma.EXAMPLES:
sage: (Magmas() & AdditiveMagmas()).Distributive() Category of distributive magmas and additive magmas sage: (Monoids() & CommutativeAdditiveGroups()).Distributive() Category of rings sage: Magmas().Distributive() Traceback (most recent call last): ... ValueError: The distributive axiom only makes sense on a magma which is simultaneously an additive magma sage: Semigroups().Distributive() Traceback (most recent call last): ... ValueError: The distributive axiom only makes sense on a magma which is simultaneously an additive magma

FinitelyGenerated
()¶ Return the subcategory of the objects of
self
that are endowed with a distinguished finite set of (multiplicative) magma generators.EXAMPLES:
This is a shorthand for
FinitelyGeneratedAsMagma()
, which see:sage: Magmas().FinitelyGenerated() Category of finitely generated magmas sage: Semigroups().FinitelyGenerated() Category of finitely generated semigroups sage: Groups().FinitelyGenerated() Category of finitely generated enumerated groups
An error is raised if this is ambiguous:
sage: (Magmas() & AdditiveMagmas()).FinitelyGenerated() Traceback (most recent call last): ... ValueError: FinitelyGenerated is ambiguous for Join of Category of magmas and Category of additive magmas. Please use explicitly one of the FinitelyGeneratedAsXXX methods
Note
Checking that there is no ambiguity currently assumes that all the other “finitely generated” axioms involve an additive structure. As of Sage 6.4, this is correct.
The use of this shorthand should be reserved for casual interactive use or when there is no risk of ambiguity.

FinitelyGeneratedAsMagma
()¶ Return the subcategory of the objects of
self
that are endowed with a distinguished finite set of (multiplicative) magma generators.A set \(S\) of elements of a multiplicative magma form a set of generators if any element of the magma can be expressed recursively from elements of \(S\) and products thereof.
It is not imposed that morphisms shall preserve the distinguished set of generators; hence this is a full subcategory.
EXAMPLES:
sage: Magmas().FinitelyGeneratedAsMagma() Category of finitely generated magmas
Being finitely generated does depend on the structure: for a ring, being finitely generated as a magma, as an additive magma, or as a ring are different concepts. Hence the name of this axiom is explicit:
sage: Rings().FinitelyGeneratedAsMagma() Category of finitely generated as magma enumerated rings
On the other hand, it does not depend on the multiplicative structure: for example a group is finitely generated if and only if it is finitely generated as a magma. A short hand is provided when there is no ambiguity, and the output tries to reflect that:
sage: Semigroups().FinitelyGenerated() Category of finitely generated semigroups sage: Groups().FinitelyGenerated() Category of finitely generated enumerated groups sage: Semigroups().FinitelyGenerated().axioms() frozenset({'Associative', 'Enumerated', 'FinitelyGeneratedAsMagma'})
Note that the set of generators may depend on the actual category; for example, in a group, one can often use less generators since it is allowed to take inverses.

JTrivial
()¶ Return the full subcategory of the \(J\)trivial objects of
self
.This axiom is in fact only meaningful for
semigroups
. This stub definition is here as a workaround for trac ticket #20515, in order to define the \(J\)trivial axiom as the intersection of the \(L\) and \(R\)trivial axioms.

Unital
()¶ Return the subcategory of the unital objects of
self
.A (multiplicative)
magma Magmas
\(M\) is unital if it admits an element \(1\), called unit, such that for all \(x\in M\),\[1 * x = x * 1 = x\]This element is necessarily unique, and should be provided as
M.one()
.EXAMPLES:
sage: Magmas().Unital() Category of unital magmas sage: Semigroups().Unital() Category of monoids sage: Monoids().Unital() Category of monoids sage: from sage.categories.associative_algebras import AssociativeAlgebras sage: AssociativeAlgebras(QQ).Unital() Category of algebras over Rational Field


class
Subquotients
(category, *args)¶ Bases:
sage.categories.subquotients.SubquotientsCategory
The category of subquotient magmas.
See
Sets.SubcategoryMethods.Subquotients()
for the general setup for subquotients. In the case of a subquotient magma \(S\) of a magma \(G\), the condition that \(r\) be a morphism inAs
can be rewritten as follows:for any two \(a,b \in S\) the identity \(a \times_S b = r(l(a) \times_G l(b))\) holds.
This is used by this category to implement the product \(\times_S\) of \(S\) from \(l\) and \(r\) and the product of \(G\).
EXAMPLES:
sage: Semigroups().Subquotients().all_super_categories() [Category of subquotients of semigroups, Category of semigroups, Category of subquotients of magmas, Category of magmas, Category of subquotients of sets, Category of sets, Category of sets with partial maps, Category of objects]

class
ParentMethods
¶ Bases:
object

product
(x, y)¶ Return the product of two elements of
self
.EXAMPLES:
sage: S = Semigroups().Subquotients().example() sage: S An example of a (sub)quotient semigroup: a quotient of the left zero semigroup sage: S.product(S(19), S(3)) 19
Here is a more elaborate example involving a sub algebra:
sage: Z = SymmetricGroup(5).algebra(QQ).center() sage: B = Z.basis() sage: B[3] * B[2] 4*B[2] + 6*B[3] + 5*B[6]


class
Unital
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton

class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory

extra_super_categories
()¶ EXAMPLES:
sage: Magmas().Commutative().Algebras(QQ).extra_super_categories() [Category of commutative magmas]
This implements the fact that the algebra of a commutative magma is commutative:
sage: Magmas().Commutative().Algebras(QQ).super_categories() [Category of magma algebras over Rational Field, Category of commutative magmas]
In particular, commutative monoid algebras are commutative algebras:
sage: Monoids().Commutative().Algebras(QQ).is_subcategory(Algebras(QQ).Commutative()) True


class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory

class
ElementMethods
¶ Bases:
object

class
ParentMethods
¶ Bases:
object

one
()¶ Return the unit of this Cartesian product.
It is built from the units for the Cartesian factors of
self
.EXAMPLES:
sage: cartesian_product([QQ, ZZ, RR]).one() (1, 1, 1.00000000000000)


extra_super_categories
()¶ Implement the fact that a Cartesian product of unital magmas is a unital magma
EXAMPLES:
sage: C = Magmas().Unital().CartesianProducts() sage: C.extra_super_categories() [Category of unital magmas] sage: C.axioms() frozenset({'Unital'}) sage: Monoids().CartesianProducts().is_subcategory(Monoids()) True

class

class
ElementMethods
¶ Bases:
object

class
Inverse
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton

class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory

extra_super_categories
()¶ Implement the fact that a Cartesian product of magmas with inverses is a magma with inverse.
EXAMPLES:
sage: C = Magmas().Unital().Inverse().CartesianProducts() sage: C.extra_super_categories() [Category of inverse unital magmas] sage: sorted(C.axioms()) ['Inverse', 'Unital']


class

class
ParentMethods
¶ Bases:
object

is_empty
()¶ Return whether
self
is empty.Since this set is a unital magma it is not empty and this method always return
False
.EXAMPLES:
sage: S = SymmetricGroup(2) sage: S.is_empty() False sage: M = Monoids().example() sage: M.is_empty() False

one
()¶ Return the unit of the monoid, that is the unique neutral element for \(*\).
Note
The default implementation is to coerce \(1\) into
self
. It is recommended to override this method because the coercion from the integers:is not always meaningful (except for \(1\));
often uses
self.one()
.
EXAMPLES:
sage: M = Monoids().example(); M An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') sage: M.one() ''


class
Realizations
(category, *args)¶

class
SubcategoryMethods
¶ Bases:
object

Inverse
()¶ Return the full subcategory of the inverse objects of
self
.An inverse :class:` (multiplicative) magma <Magmas>` is a
unital magma
such that every element admits both an inverse on the left and on the right. Such a magma is also called a loop.EXAMPLES:
sage: Magmas().Unital().Inverse() Category of inverse unital magmas sage: Monoids().Inverse() Category of groups


additional_structure
()¶ Return
self
.Indeed, the category of unital magmas defines an additional structure, namely the unit of the magma which shall be preserved by morphisms.
See also
EXAMPLES:
sage: Magmas().Unital().additional_structure() Category of unital magmas

class

super_categories
()¶ EXAMPLES:
sage: Magmas().super_categories() [Category of sets]

class