# Monoids#

class sage.categories.monoids.Monoids(base_category)#

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')

class Algebras(category, *args)#
class ElementMethods#

Bases: object

is_central()#

Return whether the element self is central.

EXAMPLES:

sage: SG4 = SymmetricGroupAlgebra(ZZ,4)                             # needs sage.groups sage.modules
sage: SG4(1).is_central()                                           # needs sage.groups sage.modules
True
sage: SG4(Permutation([1,3,2,4])).is_central()                      # needs sage.groups sage.modules
False

sage: A = GroupAlgebras(QQ).example(); A                            # needs sage.groups sage.modules
Algebra of Dihedral group of order 8
as a permutation group over Rational Field
sage: sum(A.basis()).is_central()                                   # needs sage.groups sage.modules
True

class ParentMethods#

Bases: object

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()                            # needs sage.modules
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()                          # needs sage.modules
Family (B[0], B[1], B[2], B[3], B[4], B[5], B[6], B[7], B[8], B[9], B[10], B[11])

sage: A10 = AlternatingGroup(10)                                    # needs sage.groups
sage: GroupAlgebras(QQ).example(A10).algebra_generators()           # needs sage.groups sage.modules
Family ((8,9,10), (1,2,3,4,5,6,7,8,9))

sage: A = DihedralGroup(3).algebra(QQ); A                           # needs sage.groups sage.modules
Algebra of Dihedral group of order 6 as a permutation group
over Rational Field
sage: A.algebra_generators()                                        # needs sage.groups sage.modules
Family ((1,2,3), (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: # needs sage.modules
sage: A = Monoids().example().algebra(ZZ)
sage: A.one_basis()
''
sage: A.one()
B['']
sage: A(3)
3*B['']

extra_super_categories()#

The algebra of a monoid is a bialgebra and a monoid.

EXAMPLES:

sage: C = Monoids().Algebras(QQ)
sage: C.extra_super_categories()
[Category of bialgebras over Rational Field,
Category of monoids]
sage: Monoids().Algebras(QQ).super_categories()
[Category of bialgebras with basis over Rational Field,
Category of semigroup algebras over Rational Field,
Category of unital magma algebras over Rational Field]

class CartesianProducts(category, *args)#

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 ElementMethods#

Bases: object

multiplicative_order()#

Return the multiplicative order of this element.

EXAMPLES:

sage: # needs sage.groups sage.modules
sage: G1 = SymmetricGroup(3)
sage: G2 = SL(2, 3)
sage: G = cartesian_product([G1, G2])
sage: G((G1.gen(0), G2.gen(1))).multiplicative_order()
12

class ParentMethods#

Bases: object

monoid_generators()#

Return the generators of self.

EXAMPLES:

sage: # needs sage.groups
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)                                          # needs sage.combinat
sage: C = cartesian_product([M, N])                                 # needs sage.combinat sage.groups
sage: C.monoid_generators()                                         # needs sage.combinat sage.groups
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)#

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 specifying 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)                            # needs sage.combinat
Free abelian monoid indexed by Integer Ring
sage: Monoids().Commutative().free(ZZ)                                  # needs sage.combinat
Free abelian monoid indexed by Integer Ring
sage: F.<x,y,z> = Monoids().Commutative().free(); F                     # needs sage.combinat
Free abelian monoid indexed by {'x', 'y', 'z'}

class ElementMethods#

Bases: object

inverse()#

Return the multiplicative inverse of self.

This is an alias for inversion, which can also be invoked by ~x for an element x.

Nota Bene: Element classes should implement __invert__ only.

EXAMPLES:

sage: AA(sqrt(~2)).inverse()                                            # needs sage.rings.number_field sage.symbolic
1.414213562373095?

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]])                                     # needs sage.modules
sage: A.powers(6)                                                       # needs sage.modules
[
[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 FiniteMonoids

Inverse#

alias of Groups

class ParentMethods#

Bases: object

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])                                       # needs sage.combinat
sage: M.semigroup_generators()                                          # needs sage.combinat
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                                  # needs sage.combinat
A submonoid of (Ring of integers modulo 15) with 2 generators
sage: M.list()                                                          # needs sage.combinat
[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                               # needs sage.combinat
A subsemigroup of (Ring of integers modulo 15) with 2 generators
sage: S.list()                                                          # needs sage.combinat
[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())                    # needs sage.combinat
sage: M2 is M                                                           # needs sage.combinat
True

class Subquotients(category, *args)#
class ParentMethods#

Bases: object

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)#
class ParentMethods#

Bases: object

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                    # needs sage.modules
The subset algebra of {1, 2, 3} over Rational Field
sage: A.one.__module__                                              # needs sage.modules
'sage.categories.monoids'
sage: A.one()                                                       # needs sage.modules
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 specifying 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)                                            # needs sage.combinat
Free monoid indexed by Integer Ring
sage: Monoids().free(ZZ)                                                    # needs sage.combinat
Free monoid indexed by Integer Ring
sage: F.<x,y,z> = Monoids().free(); F                                       # needs sage.combinat
Free monoid indexed by {'x', 'y', 'z'}