Examples of semigroups in cython

class sage.categories.examples.semigroups_cython.IdempotentSemigroups

Bases: Category

class ElementMethods

Bases: object

is_idempotent()

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.categories.examples.semigroups_cython import LeftZeroSemigroup
>>> S = LeftZeroSemigroup()
>>> S(Integer(2)).is_idempotent()
True

super_categories()

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.categories.examples.semigroups_cython import IdempotentSemigroups
>>> IdempotentSemigroups().super_categories()
[Category of semigroups]

class sage.categories.examples.semigroups_cython.LeftZeroSemigroup

Bases: 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

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

Python

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

This is the semigroup which contains all sort of objects:

Sage

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

Python

>>> from sage.all import *
>>> S.some_elements()
[3, 42, 'a', 3.4, 'raton laveur']

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

Sage

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_construction() . . . 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

Python

>>> from sage.all import *
>>> S('hello') * S('world')
'hello'

>>> S(Integer(3))*S(Integer(1))*S(Integer(2))
3

>>> S(Integer(3))**Integer(12312321312321)
3

>>> TestSuite(S).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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

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

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

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

Python

>>> from sage.all import *
>>> S(Integer(42)).is_idempotent
<bound method IdempotentSemigroups.ElementMethods.is_idempotent of 42>
>>> S(Integer(42)).is_idempotent()
True

>>> S(Integer(42))._pow_int
<bound method IdempotentSemigroups.ElementMethods._pow_int of 42>
>>> S(Integer(42))**Integer(10)
42

>>> S(Integer(42)).is_idempotent
<bound method IdempotentSemigroups.ElementMethods.is_idempotent of 42>
>>> S(Integer(42)).is_idempotent()
True

Element

alias of LeftZeroSemigroupElement

class sage.categories.examples.semigroups_cython.LeftZeroSemigroupElement

Bases: Element

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.categories.examples.semigroups_cython import LeftZeroSemigroup
>>> S = LeftZeroSemigroup()
>>> x = S(Integer(3))
>>> TestSuite(x).run()