Examples of semigroups in cython

class sage.categories.examples.semigroups_cython.IdempotentSemigroups(s=None)

Bases: sage.categories.category.Category

class ElementMethods

Bases: object

is_idempotent()

EXAMPLES:

sage: from sage.categories.examples.semigroups_cython import LeftZeroSemigroup
sage: S = LeftZeroSemigroup()
sage: S(2).is_idempotent()
True
super_categories()

EXAMPLES:

sage: from sage.categories.examples.semigroups_cython import IdempotentSemigroups
sage: IdempotentSemigroups().super_categories()
[Category of semigroups]
class sage.categories.examples.semigroups_cython.LeftZeroSemigroup

Bases: sage.categories.examples.semigroups.LeftZeroSemigroup

An example of semigroup

This class illustrates a minimal implementation of a semi-group where the element class is an extension type, and still gets code from the category. The category itself must be a Python class though.

This is purely a proof of concept. The code obviously needs refactorisation!

Comments:

  • one cannot play ugly class surgery tricks (as with _mul_parent). available operations should really be declared to the coercion model!

EXAMPLES:

sage: from sage.categories.examples.semigroups_cython import LeftZeroSemigroup
sage: S = LeftZeroSemigroup(); S
An example of a semigroup: the left zero semigroup

This is the semigroup which contains all sort of objects:

sage: S.some_elements()
[3, 42, 'a', 3.4, 'raton laveur']

with product rule is given by \(a \times b = a\) for all \(a,b\).

sage: S('hello') * S('world')
'hello'

sage: S(3)*S(1)*S(2)
3

sage: S(3)^12312321312321
3

sage: TestSuite(S).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_new() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass

That’s really the only method which is obtained from the category …

sage: S(42).is_idempotent
<bound method IdempotentSemigroups.element_class.is_idempotent of 42>
sage: S(42).is_idempotent()
True

sage: S(42)._pow_int
<bound method IdempotentSemigroups.element_class._pow_int of 42>
sage: S(42)^10
42

sage: S(42).is_idempotent
<bound method IdempotentSemigroups.element_class.is_idempotent of 42>
sage: S(42).is_idempotent()
True
Element

alias of LeftZeroSemigroupElement

class sage.categories.examples.semigroups_cython.LeftZeroSemigroupElement

Bases: sage.structure.element.Element

EXAMPLES:

sage: from sage.categories.examples.semigroups_cython import LeftZeroSemigroup
sage: S = LeftZeroSemigroup()
sage: x = S(3)
sage: TestSuite(x).run()