Trac #17160: Finitely generated axiom for (mutiplicative) magmas, semigroups, monoids, groups

This introduce an axiom FinitelyGeneratedAsMagma, as well as related
categories with axioms for magmas, semigroups and groups::
{{{
    sage: Groups().FinitelyGeneratedAsMagma()
    Category of finitely generated groups
}}}
For ease of notations, when there is no ambiguity, one can do::
{{{
    sage: Groups().FinitelyGenerated()
    Category of finitely generated groups
}}}

One motivation for this change (for #8678) is that finite semigroups
in Sage used to be automatically endowed with an `EnumeratedSets`
structure; the default enumeration is then obtained by iteratively
multiplying the semigroup generators. This forced any finite semigroup
to either implement an enumeration, or provide semigroup generators;
this was often inconvenient.

Instead, finite semigroups that provide a distinguished finite set of
generators with `semigroup_generators` should now explicitly declare
themselves in the category of `FinitelyGeneratedSemigroups`:
{{{
    sage: Semigroups().FinitelyGenerated()
    Category of finitely generated semigroups
}}}
This is a backward incompatible change.

TODO:
- Use the occasion to migrate TransitiveIdeal to
RecursivelyEnumeratedSet

URL: http://trac.sagemath.org/17160
Reported by: nthiery
Ticket author(s): Nicolas M. Thiéry
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -461,7 +461,7 @@ And indeed, ``MS2`` has *more* methods than ``MS1``::
     sage: len([s for s in dir(MS1) if inspect.ismethod(getattr(MS1,s,None))])
     57
     sage: len([s for s in dir(MS2) if inspect.ismethod(getattr(MS2,s,None))])
-    82
+    83
 
 This is because the class of ``MS2`` also inherits from the parent
 class for algebras::

src/sage/categories/category.py

@@ -2625,7 +2625,7 @@ def category_graph(categories = None):
         Graphics object consisting of 20 graphics primitives
 
         sage: sage.categories.category.category_graph().plot()
-        Graphics object consisting of 310 graphics primitives
+        Graphics object consisting of ... graphics primitives
     """
     from sage import graphs
     if categories is None:

src/sage/categories/category_with_axiom.py

@@ -1675,13 +1675,14 @@ class ``Sets.Finite``), or in a separate file (typically in a class
 
 all_axioms = AxiomContainer()
 all_axioms += ("Flying", "Blue",
-              "Facade", "Finite", "Infinite",
-              "FiniteDimensional", "Connected", "WithBasis",
-              "Irreducible",
-              "Commutative", "Associative", "Inverse", "Unital", "Division", "NoZeroDivisors",
-              "AdditiveCommutative", "AdditiveAssociative", "AdditiveInverse", "AdditiveUnital",
-              "Distributive",
-              "Endset"
+               "FinitelyGeneratedAsMagma",
+               "Facade", "Finite", "Infinite",
+               "FiniteDimensional", "Connected", "WithBasis",
+               "Irreducible",
+               "Commutative", "Associative", "Inverse", "Unital", "Division", "NoZeroDivisors",
+               "AdditiveCommutative", "AdditiveAssociative", "AdditiveInverse", "AdditiveUnital",
+               "Distributive",
+               "Endset"
               )
 
 def uncamelcase(s,separator=" "):
@@ -2129,25 +2130,27 @@ def super_categories(self):
         ``self``, as per :meth:`Category.super_categories`.
 
         This implements the property that if ``As`` is a subcategory
-        of ``Bs``, then the intersection of As with ``FiniteSets()``
+        of ``Bs``, then the intersection of ``As`` with ``FiniteSets()``
         is a subcategory of ``As`` and of the intersection of ``Bs``
         with ``FiniteSets()``.
 
-        EXAMPLES::
-
-            sage: FiniteSets().super_categories()
-            [Category of sets]
-
-            sage: FiniteSemigroups().super_categories()
-            [Category of semigroups, Category of finite enumerated sets]
-
         EXAMPLES:
 
         A finite magma is both a magma and a finite set::
 
             sage: Magmas().Finite().super_categories()
             [Category of magmas, Category of finite sets]
 
+        Variants::
+
+            sage: Sets().Finite().super_categories()
+            [Category of sets]
+
+            sage: Monoids().Finite().super_categories()
+            [Category of monoids, Category of finite semigroups]
+
+        EXAMPLES:
+
         TESTS::
 
             sage: from sage.categories.category_with_axiom import TestObjects
@@ -2232,7 +2235,12 @@ def _repr_object_names_static(category, axioms):
             sage: from sage.categories.homsets import Homsets
             sage: CategoryWithAxiom._repr_object_names_static(Homsets(), ["Endset"])
             'endsets'
+            sage: CategoryWithAxiom._repr_object_names_static(PermutationGroups(), ["FinitelyGeneratedAsMagma"])
+            'finitely generated permutation groups'
+            sage: CategoryWithAxiom._repr_object_names_static(Rings(), ["FinitelyGeneratedAsMagma"])
+            'finitely generated as magma rings'
         """
+        from sage.categories.additive_magmas import AdditiveMagmas
         axioms = canonicalize_axioms(all_axioms,axioms)
         base_category = category._without_axioms(named=True)
         if isinstance(base_category, CategoryWithAxiom): # Smelly runtime type checking
@@ -2254,6 +2262,9 @@ def _repr_object_names_static(category, axioms):
             elif axiom == "Endset" and "homsets" in result:
                 # Without the space at the end to handle Homsets().Endset()
                 result = result.replace("homsets", "endsets", 1)
+            elif axiom == "FinitelyGeneratedAsMagma" and \
+                 not base_category.is_subcategory(AdditiveMagmas()):
+                result = "finitely generated " + result
             else:
                 result = uncamelcase(axiom) + " " + result
         return result

src/sage/categories/coxeter_groups.py

@@ -39,7 +39,7 @@ class CoxeterGroups(Category_singleton):
         sage: C = CoxeterGroups(); C
         Category of coxeter groups
         sage: C.super_categories()
-        [Category of groups, Category of enumerated sets]
+        [Category of finitely generated groups]
 
         sage: W = C.example(); W
         The symmetric group on {0, ..., 3}
@@ -117,9 +117,9 @@ def super_categories(self):
         EXAMPLES::
 
             sage: CoxeterGroups().super_categories()
-            [Category of groups, Category of enumerated sets]
+            [Category of finitely generated groups]
         """
-        return [Groups(), EnumeratedSets()]
+        return [Groups().FinitelyGenerated()]
 
     Finite = LazyImport('sage.categories.finite_coxeter_groups', 'FiniteCoxeterGroups')
     Algebras = LazyImport('sage.categories.coxeter_group_algebras', 'CoxeterGroupAlgebras')

src/sage/categories/examples/finite_monoids.py

@@ -13,7 +13,7 @@
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
 from sage.structure.element_wrapper import ElementWrapper
-from sage.categories.all import FiniteMonoids
+from sage.categories.all import Monoids
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 
@@ -29,7 +29,7 @@ class IntegerModMonoid(UniqueRepresentation, Parent):
         An example of a finite multiplicative monoid: the integers modulo 12
 
         sage: S.category()
-        Category of finite monoids
+        Category of finitely generated finite monoids
 
     We conclude by running systematic tests on this monoid::
 
@@ -72,7 +72,7 @@ def __init__(self, n = 12):
 
         """
         self.n = n
-        Parent.__init__(self, category = FiniteMonoids())
+        Parent.__init__(self, category = Monoids().Finite().FinitelyGenerated())
 
     def _repr_(self):
         r"""

src/sage/categories/examples/finite_semigroups.py

@@ -10,7 +10,7 @@
 
 from sage.misc.cachefunc import cached_method
 from sage.sets.family import Family
-from sage.categories.all import FiniteSemigroups
+from sage.categories.semigroups import Semigroups
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
 from sage.structure.element_wrapper import ElementWrapper
@@ -114,7 +114,7 @@ def __init__(self, alphabet=('a','b','c','d')):
             sage: TestSuite(S).run()
         """
         self.alphabet = alphabet
-        Parent.__init__(self, category = FiniteSemigroups())
+        Parent.__init__(self, category = Semigroups().Finite().FinitelyGenerated())
 
     def _repr_(self):
         r"""

src/sage/categories/examples/semigroups.py

@@ -173,25 +173,7 @@ class FreeSemigroup(UniqueRepresentation, Parent):
 
     TESTS::
 
-        sage: TestSuite(S).run(verbose = True)
-        running ._test_an_element() . . . pass
-        running ._test_associativity() . . . 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_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_not_implemented_methods() . . . pass
-        running ._test_pickling() . . . pass
-        running ._test_some_elements() . . . pass
+        sage: TestSuite(S).run()
     """
     def __init__(self, alphabet=('a','b','c','d')):
         r"""
@@ -214,7 +196,7 @@ def __init__(self, alphabet=('a','b','c','d')):
 
         """
         self.alphabet = alphabet
-        Parent.__init__(self, category = Semigroups())
+        Parent.__init__(self, category = Semigroups().FinitelyGenerated())
 
     def _repr_(self):
         r"""

src/sage/categories/finite_coxeter_groups.py

@@ -23,7 +23,9 @@ class FiniteCoxeterGroups(CategoryWithAxiom):
         sage: FiniteCoxeterGroups()
         Category of finite coxeter groups
         sage: FiniteCoxeterGroups().super_categories()
-        [Category of finite groups, Category of coxeter groups]
+        [Category of coxeter groups,
+         Category of finite groups,
+         Category of finite finitely generated semigroups]
 
         sage: G = FiniteCoxeterGroups().example()
         sage: G.cayley_graph(side = "right").plot()

src/sage/categories/finite_fields.py

@@ -12,21 +12,24 @@
 #******************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
+from sage.categories.enumerated_sets import EnumeratedSets
 
 class FiniteFields(CategoryWithAxiom):
     """
     The category of finite fields.
 
     EXAMPLES::
 
-        sage: K = FiniteFields()
-        sage: K
+        sage: K = FiniteFields(); K
         Category of finite fields
 
-    A finite field is a finite monoid with the structure of a field::
+    A finite field is a finite monoid with the structure of a field;
+    it is currently assumed to be enumerated::
 
         sage: K.super_categories()
-        [Category of fields, Category of finite commutative rings]
+        [Category of fields,
+         Category of finite commutative rings,
+         Category of finite enumerated sets]
 
     Some examples of membership testing and coercion::
 
@@ -41,11 +44,24 @@ class FiniteFields(CategoryWithAxiom):
 
     TESTS::
 
-        sage: TestSuite(FiniteFields()).run()
-        sage: FiniteFields().is_subcategory(FiniteEnumeratedSets())
+        sage: K is Fields().Finite()
         True
+        sage: TestSuite(K).run()
     """
 
+    def extra_super_categories(self):
+        r"""
+        Any finite field is assumed to be endowed with an enumeration.
+
+        TESTS::
+
+            sage: Fields().Finite().extra_super_categories()
+            [Category of finite enumerated sets]
+            sage: FiniteFields().is_subcategory(FiniteEnumeratedSets())
+            True
+        """
+        return [EnumeratedSets().Finite()]
+
     def __contains__(self, x):
         """
         EXAMPLES::

src/sage/categories/finite_permutation_groups.py

@@ -9,26 +9,36 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.categories.magmas import Magmas
 from sage.categories.category_with_axiom import CategoryWithAxiom
+from sage.categories.permutation_groups import PermutationGroups
 
 class FinitePermutationGroups(CategoryWithAxiom):
     r"""
     The category of finite permutation groups, i.e. groups concretely
     represented as groups of permutations acting on a finite set.
 
+    It is currently assumed that any finite permutation group comes
+    endowed with a distinguished finite set of generators (method
+    ``group_generators``); this is the case for all the existing
+    implementations in Sage.
+
     EXAMPLES::
 
-        sage: FinitePermutationGroups()
+        sage: C = PermutationGroups().Finite(); C
         Category of finite permutation groups
-        sage: FinitePermutationGroups().super_categories()
-        [Category of permutation groups, Category of finite groups]
+        sage: C.super_categories()
+        [Category of permutation groups,
+         Category of finite groups,
+         Category of finite finitely generated semigroups]
 
-        sage: FinitePermutationGroups().example()
+        sage: C.example()
         Dihedral group of order 6 as a permutation group
 
     TESTS::
 
-        sage: C = FinitePermutationGroups()
+        sage: C is FinitePermutationGroups()
+        True
         sage: TestSuite(C).run()
 
         sage: G = FinitePermutationGroups().example()
@@ -72,6 +82,17 @@ def example(self):
         from sage.groups.perm_gps.permgroup_named import DihedralGroup
         return DihedralGroup(3)
 
+    def extra_super_categories(self):
+        """
+        Any permutation group is assumed to be endowed with a finite set of generators.
+
+        TESTS:
+
+            sage: PermutationGroups().Finite().extra_super_categories()
+            [Category of finitely generated magmas]
+        """
+        return [Magmas().FinitelyGenerated()]
+
     class ParentMethods:
         # TODO
         #  - Port features from MuPAD-Combinat, lib/DOMAINS/CATEGORIES/PermutationGroup.mu