Monoids

class sage.categories.monoids.Monoids(s=None)

Bases: sage.categories.category.Category

The category of (multiplicative) monoids, i.e. semigroups with a unit.

EXAMPLES:

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

TESTS:

sage: C = Monoids()
sage: TestSuite(C).run()
class ElementMethods
is_one()

Returns whether self is the one of the monoid

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

TESTS:

sage: S = Monoids().example()
sage: S.one().is_one()
True
sage: S("aa").is_one()
False
class Monoids.ParentMethods
one(*args, **kwds)

Returns the one of the monoid, that is the unique neutral element for *.

EXAMPLES:

sage: M = Monoids().example(); M
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
sage: M.one()
''
one_element()

Backward compatibility alias for self.one().

TESTS:

sage: S = Monoids().example()
sage: S.one_element()
''
prod(args)

n-ary product

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’
Monoids.super_categories(*args, **kwds)

Returns a list of the immediate super categories of self.

EXAMPLES:

sage: Monoids().super_categories()
[Category of semigroups]

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