Monoids

class sage.categories.monoids.Monoids(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of (multiplicative) monoids.

A monoid is a unital semigroup, that is a set endowed with a multiplicative binary operation \(*\) which is associative and admits a unit (see Wikipedia article Monoid).

EXAMPLES:

sage: Monoids()
Category of monoids
sage: Monoids().super_categories()
[Category of semigroups, Category of unital magmas]
sage: Monoids().all_super_categories()
[Category of monoids,
 Category of semigroups,
 Category of unital magmas, Category of magmas,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]

sage: Monoids().axioms()
frozenset({'Associative', 'Unital'})
sage: Semigroups().Unital()
Category of monoids

sage: Monoids().example()
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
sage: S = Monoids().example()
sage: x = S("aa")
sage: x^0, x^1, x^2, x^3, x^4, x^5
('', 'aa', 'aaaa', 'aaaaaa', 'aaaaaaaa', 'aaaaaaaaaa')
class Algebras(category, *args)

Bases: sage.categories.algebra_functor.AlgebrasCategory

class ElementMethods
is_central()

Return whether the element self is central.

EXAMPLES:

sage: SG4 = SymmetricGroupAlgebra(ZZ,4)
sage: SG4(1).is_central()
True
sage: SG4(Permutation([1,3,2,4])).is_central()
False
sage: A = GroupAlgebras(QQ).example(); A
Algebra of Dihedral group of order 8 as a permutation group over Rational Field
sage: sum(i for i in A.basis()).is_central()
True
class ParentMethods
algebra_generators()

Return generators for this algebra.

For a monoid algebra, the algebra generators are built from the monoid generators if available and from the semigroup generators otherwise.

EXAMPLES:

sage: M = Monoids().example(); M
An example of a monoid:
the free monoid generated by ('a', 'b', 'c', 'd')
sage: M.monoid_generators()
Finite family {'a': 'a', 'b': 'b', 'c': 'c', 'd': 'd'}
sage: M.algebra(ZZ).algebra_generators()
Finite family {'a': B['a'], 'b': B['b'], 'c': B['c'], 'd': B['d']}

sage: Z12 = Monoids().Finite().example(); Z12
An example of a finite multiplicative monoid:
the integers modulo 12
sage: Z12.monoid_generators()
Traceback (most recent call last):
...
AttributeError: 'IntegerModMonoid_with_category' object
has no attribute 'monoid_generators'
sage: Z12.semigroup_generators()
Family (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
sage: Z12.algebra(QQ).algebra_generators()
Finite family {0: B[0], 1: B[1], 2: B[2], 3: B[3],  4: B[4],   5: B[5],
               6: B[6], 7: B[7], 8: B[8], 9: B[9], 10: B[10], 11: B[11]}


sage: GroupAlgebras(QQ).example(AlternatingGroup(10)).algebra_generators()
Finite family {0: (8,9,10), 1: (1,2,3,4,5,6,7,8,9)}

sage: A = DihedralGroup(3).algebra(QQ); A
Algebra of Dihedral group of order 6 as a permutation group
 over Rational Field
sage: A.algebra_generators()
Finite family {0: (1,2,3), 1: (1,3)}
one_basis()

Return the unit of the monoid, which indexes the unit of this algebra, as per AlgebrasWithBasis.ParentMethods.one_basis().

EXAMPLES:

sage: A = Monoids().example().algebra(ZZ)
sage: A.one_basis()
''
sage: A.one()
B['']
sage: A(3)
3*B['']
extra_super_categories()

EXAMPLES:

sage: Monoids().Algebras(QQ).extra_super_categories()
[Category of monoids]
sage: Monoids().Algebras(QQ).super_categories()
[Category of algebras with basis over Rational Field,
 Category of semigroup algebras over Rational Field,
 Category of unital magma algebras over Rational Field]
class CartesianProducts(category, *args)

Bases: sage.categories.cartesian_product.CartesianProductsCategory

The category of monoids constructed as Cartesian products of monoids.

This construction gives the direct product of monoids. See Wikipedia article Direct_product for more information.

class ParentMethods
monoid_generators()

Return the generators of self.

EXAMPLES:

sage: M = Monoids.free([1,2,3])
sage: N = Monoids.free(['a','b'])
sage: C = cartesian_product([M, N])
sage: C.monoid_generators()
Family ((F[1], 1), (F[2], 1), (F[3], 1),
        (1, F['a']), (1, F['b']))

An example with an infinitely generated group (a better output is needed):

sage: N = Monoids.free(ZZ)
sage: C = cartesian_product([M, N])
sage: C.monoid_generators()
Lazy family (gen(i))_{i in The Cartesian product of (...)}
extra_super_categories()

A Cartesian product of monoids is endowed with a natural group structure.

EXAMPLES:

sage: C = Monoids().CartesianProducts()
sage: C.extra_super_categories()
[Category of monoids]
sage: sorted(C.super_categories(), key=str)
[Category of Cartesian products of semigroups,
 Category of Cartesian products of unital magmas,
 Category of monoids]
class Commutative(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

Category of commutative (abelian) monoids.

A monoid \(M\) is commutative if \(xy = yx\) for all \(x,y \in M\).

static free(index_set=None, names=None, **kwds)

Return a free abelian monoid on \(n\) generators or with the generators indexed by a set \(I\).

A free monoid is constructed by specifing either:

  • the number of generators and/or the names of the generators, or
  • the indexing set for the generators.

INPUT:

  • index_set – (optional) an index set for the generators; if an integer, then this represents \(\{0, 1, \ldots, n-1\}\)
  • names – a string or list/tuple/iterable of strings (default: 'x'); the generator names or name prefix

EXAMPLES:

sage: Monoids.Commutative.free(index_set=ZZ)
Free abelian monoid indexed by Integer Ring
sage: Monoids().Commutative().free(ZZ)
Free abelian monoid indexed by Integer Ring
sage: F.<x,y,z> = Monoids().Commutative().free(); F
Free abelian monoid indexed by {'x', 'y', 'z'}
class ElementMethods
is_one()

Return whether self is the one of the monoid.

The default implementation is to compare with self.one().

powers(n)

Return the list \([x^0, x^1, \ldots, x^{n-1}]\).

EXAMPLES:

sage: A = Matrix([[1, 1], [-1, 0]])
sage: A.powers(6)
[
[1 0]  [ 1  1]  [ 0  1]  [-1  0]  [-1 -1]  [ 0 -1]
[0 1], [-1  0], [-1 -1], [ 0 -1], [ 1  0], [ 1  1]
]
Finite

alias of sage.categories.finite_monoids.FiniteMonoids

Inverse

alias of sage.categories.groups.Groups

class ParentMethods
prod(args)

n-ary product of elements of self.

INPUT:

  • args – a list (or iterable) of elements of self

Returns the product of the elements in args, as an element of self.

EXAMPLES:

sage: S = Monoids().example()
sage: S.prod([S('a'), S('b')])
'ab'
semigroup_generators()

Return the generators of self as a semigroup.

The generators of a monoid \(M\) as a semigroup are the generators of \(M\) as a monoid and the unit.

EXAMPLES:

sage: M = Monoids().free([1,2,3])
sage: M.semigroup_generators()
Family (1, F[1], F[2], F[3])
submonoid(generators, category=None)

Return the multiplicative submonoid generated by generators.

INPUT:

  • generators – a finite family of elements of self, or a list, iterable, … that can be converted into one (see Family).
  • category – a category

This is a shorthand for Semigroups.ParentMethods.subsemigroup() that specifies that this is a submonoid, and in particular that the unit is self.one().

EXAMPLES:

sage: R = IntegerModRing(15)
sage: M = R.submonoid([R(3),R(5)]); M
A submonoid of (Ring of integers modulo 15) with 2 generators
sage: M.list()
[1, 3, 5, 9, 0, 10, 12, 6]

Not the presence of the unit, unlike in:

sage: S = R.subsemigroup([R(3),R(5)]); S
A subsemigroup of (Ring of integers modulo 15) with 2 generators
sage: S.list()
[3, 5, 9, 0, 10, 12, 6]

This method is really a shorthand for subsemigroup:

sage: M2 = R.subsemigroup([R(3),R(5)], one=R.one())
sage: M2 is M
True
class Subquotients(category, *args)

Bases: sage.categories.subquotients.SubquotientsCategory

class ParentMethods
one()

Returns the multiplicative unit of this monoid, obtained by retracting that of the ambient monoid.

EXAMPLES:

sage: S = Monoids().Subquotients().example() # todo: not implemented
sage: S.one()                                # todo: not implemented
class WithRealizations(category, *args)

Bases: sage.categories.with_realizations.WithRealizationsCategory

class ParentMethods
one()

Return the unit of this monoid.

This default implementation returns the unit of the realization of self given by a_realization().

EXAMPLES:

sage: A = Sets().WithRealizations().example(); A
The subset algebra of {1, 2, 3} over Rational Field
sage: A.one.__module__
'sage.categories.monoids'
sage: A.one()
F[{}]
static free(index_set=None, names=None, **kwds)

Return a free monoid on \(n\) generators or with the generators indexed by a set \(I\).

A free monoid is constructed by specifing either:

  • the number of generators and/or the names of the generators
  • the indexing set for the generators

INPUT:

  • index_set – (optional) an index set for the generators; if an integer, then this represents \(\{0, 1, \ldots, n-1\}\)
  • names – a string or list/tuple/iterable of strings (default: 'x'); the generator names or name prefix

EXAMPLES:

sage: Monoids.free(index_set=ZZ)
Free monoid indexed by Integer Ring
sage: Monoids().free(ZZ)
Free monoid indexed by Integer Ring
sage: F.<x,y,z> = Monoids().free(); F
Free monoid indexed by {'x', 'y', 'z'}