gh-37258: add type annotations to gens methods

    
This is adding type annotations to many `gens` methods.

This will help to make them all return `tuple` later.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: #37258
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe

src/sage/categories/examples/finite_dimensional_lie_algebras_with_basis.py

@@ -242,7 +242,7 @@ def basis(self):
 
     lie_algebra_generators = basis
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self``.
 

src/sage/categories/finite_dimensional_modules_with_basis.py

@@ -36,7 +36,7 @@ class FiniteDimensionalModulesWithBasis(CategoryWithAxiom_over_base_ring):
 
     class ParentMethods:
 
-        def gens(self):
+        def gens(self) -> tuple:
             """
             Return the generators of ``self``.
 

src/sage/categories/pushout.py

@@ -3797,7 +3797,7 @@ def __call__(self, R):
         return PermutationGroup([g for g in (R.gens() + self.gens()) if not g.is_one()],
                                 domain=self._domain)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         EXAMPLES::
 

src/sage/categories/semigroups.py

@@ -906,7 +906,7 @@ def algebra_generators(self):
             # gens / monoid/group/*_generators, these methods could possibly
             # be removed in favor of aliases gens -> xxx_generators in
             # the Algebras.FinitelyGenerated hierarchy
-            def gens(self):
+            def gens(self) -> tuple:
                 r"""
                 Return the generators of ``self``.
 

src/sage/coding/linear_code_no_metric.py

@@ -373,7 +373,7 @@ def rate(self):
         return self.dimension() / self.length()
 
     @cached_method
-    def gens(self):
+    def gens(self) -> list:
         r"""
         Return the generators of this code as a list of vectors.
 

src/sage/combinat/colored_permutations.py

@@ -651,7 +651,7 @@ def _inverse_simple_reflections(self):
         return {i: ~s[i] for i in self.index_set()}
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self``.
 

src/sage/combinat/free_dendriform_algebra.py

@@ -285,7 +285,7 @@ def change_ring(self, R):
         """
         return FreeDendriformAlgebra(R, names=self.variable_names())
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self`` (as an algebra).
 

src/sage/combinat/free_prelie_algebra.py

@@ -348,7 +348,7 @@ def change_ring(self, R):
         """
         return FreePreLieAlgebra(R, names=self.variable_names())
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self`` (as an algebra).
 

src/sage/geometry/hyperplane_arrangement/arrangement.py

@@ -3851,7 +3851,7 @@ def ngens(self):
         return len(self._names)
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the coordinate hyperplanes.
 

src/sage/geometry/linear_expression.py

@@ -534,7 +534,7 @@ def ngens(self):
         return len(self._names)
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self``.
 

src/sage/geometry/toric_lattice.py

@@ -1511,7 +1511,7 @@ def __init__(self, V, W, check=True, positive_point=None, positive_dual_point=No
             raise ValueError('You may not specify both positive_point and positive_dual_point.')
         self._flip_sign_of_generator = (scalar_product < 0)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of the quotient.
 

src/sage/groups/abelian_gps/abelian_group.py

@@ -928,7 +928,7 @@ def gen(self, i=0):
             x[i] = 1
         return self.element_class(self, x)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of the group.
 
@@ -1875,7 +1875,7 @@ def _repr_(self):
         s += '{' + ', '.join(map(str, self.gens())) + '}'
         return s
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators for this subgroup.
 

src/sage/groups/abelian_gps/dual_abelian_group.py

@@ -267,7 +267,7 @@ def gen(self, i=0):
             x[i] = 1
         return self.element_class(self, x)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators for the group.
 

src/sage/groups/additive_abelian/additive_abelian_group.py

@@ -416,7 +416,7 @@ def __init__(self, cover, rels, gens):
         AdditiveAbelianGroup_class.__init__(self, cover, rels)
         self._orig_gens = tuple(self(x) for x in gens)
 
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the specified generators for self (as a tuple). Compare
         ``self.smithform_gens()``.

src/sage/groups/cactus_group.py

@@ -218,7 +218,7 @@ def group_generators(self):
         return Family(l, lambda x: self.element_class(self, [x]))
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self`` as a tuple.
 
@@ -943,7 +943,7 @@ def gen(self, i):
         return self.gens()[i]
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the generators of ``self``.
 

src/sage/groups/kernel_subgroup.py

@@ -69,7 +69,7 @@ def _repr_(self):
         """
         return "Kernel subgroup defined by {}".format(self._morphism)
 
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the generators of ``self``.
 

src/sage/groups/lie_gps/nilpotent_lie_group.py

@@ -310,7 +310,7 @@ def _dRx(self):
         return R_a.differential(self.one()).matrix().subs(asubs)
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return a tuple of elements whose one-parameter subgroups generate
         the Lie group.

src/sage/groups/matrix_gps/finitely_generated.py

@@ -377,7 +377,7 @@ def __init__(self, degree, base_ring, generator_matrices, category=None):
         MatrixGroup_generic.__init__(self, degree, base_ring, category=category)
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of the matrix group.
 

src/sage/groups/raag.py

@@ -817,7 +817,7 @@ def algebra_generators(self):
         from sage.sets.family import Family
         return Family(V, lambda x: d[x])
 
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return the generators of ``self`` (as an algebra).
 

src/sage/libs/eclib/interface.py

@@ -569,7 +569,7 @@ def saturate(self, bound=-1, lower=2):
             self.__two_descent_data().saturate(bound, lower)
             self.__saturate = bound
 
-    def gens(self):
+    def gens(self) -> list:
         """
         Return a list of the generators for the Mordell-Weil group.
 

src/sage/modular/arithgroup/arithgroup_generic.py

@@ -1028,7 +1028,7 @@ def generators(self, algorithm="farey"):
         else:
             raise ValueError("Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm)
 
-    def gens(self, *args, **kwds):
+    def gens(self, *args, **kwds) -> tuple:
         r"""
         Return a tuple of generators for this congruence subgroup.
 

src/sage/modules/fg_pid/fgp_module.py

@@ -1017,7 +1017,7 @@ def invariants(self, include_ones=False):
         self.invariants.set_cache(w, False)
         return self.invariants(include_ones)
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return tuple of elements `g_0,...,g_n` of ``self`` such that the module generated by
         the `g_i` is isomorphic to the direct sum of `R/e_i R`, where `e_i` are the

src/sage/modules/free_module.py

@@ -2515,7 +2515,7 @@ def cardinality(self):
             return sage.rings.integer.Integer(1)
         return self.base_ring().cardinality() ** self.rank()
 
-    __len__ = cardinality # for backward compatibility
+    __len__ = cardinality  # for backward compatibility
 
     def basis(self):
         """
@@ -2532,7 +2532,7 @@ def basis(self):
         """
         raise NotImplementedError
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return a tuple of basis elements of ``self``.
 

src/sage/modules/quotient_module.py

@@ -119,7 +119,7 @@ def __hash__(self):
         """
         return self.__hash
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of this module.
 
@@ -132,7 +132,7 @@ def gens(self):
             sage: Q.gens()
             ((1, 0), (0, 1))
         """
-        return tuple([self(list(g)) for g in self._module.gens()])
+        return tuple(self(list(g)) for g in self._module.gens())
 
     def gen(self, i=0):
         """

src/sage/modules/submodule.py

@@ -194,7 +194,7 @@ def relations(self):
         """
         return self._ambient.relations()
 
-    def gens(self):
+    def gens(self) -> list:
         """
         Return the generators of this submodule.
 

src/sage/modules/torsion_quadratic_module.py

@@ -520,7 +520,7 @@ def gram_matrix_quadratic(self):
             G[i, i] = gens[i].q().lift()
         return G
 
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return generators of ``self``.
 

src/sage/monoids/free_abelian_monoid.py

@@ -265,12 +265,12 @@ def gen(self, i=0):
         n = self.__ngens
         if i < 0 or not i < n:
             raise IndexError(f"argument i (= {i}) must be between 0 and {n-1}")
-        x = [ 0 for j in range(n) ]
+        x = [0 for j in range(n)]
         x[int(i)] = 1
         return self.element_class(self, x)
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of ``self``.
 
@@ -280,7 +280,7 @@ def gens(self):
             sage: F.gens()
             (a0, a1, a2, a3, a4)
         """
-        return tuple([self.gen(i) for i in range(self.__ngens)])
+        return tuple(self.gen(i) for i in range(self.__ngens))
 
     def ngens(self):
         """

src/sage/monoids/monoid.py

@@ -46,9 +46,9 @@ def __init__(self, names):
         Parent.__init__(self, base=self, names=names, category=category)
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
-        Returns the generators for ``self``.
+        Return the generators for ``self``.
 
         EXAMPLES::
 
@@ -60,7 +60,7 @@ def gens(self):
 
     def monoid_generators(self):
         r"""
-        Returns the generators for ``self``.
+        Return the generators for ``self``.
 
         EXAMPLES::
 

src/sage/quadratic_forms/bqf_class_group.py

@@ -314,7 +314,7 @@ def abelian_group(self):
         gens = [BinaryQF(g) for g in gens]
         return AdditiveAbelianGroupWrapper(self, gens, ords)
 
-    def gens(self):
+    def gens(self) -> list:
         r"""
         Return a generating set of this form class group.
 

src/sage/quivers/algebra.py

@@ -172,7 +172,7 @@ def __init__(self, k, P, order="negdegrevlex"):
                          bracket=False)
         self._assign_names(self._semigroup.variable_names())
 
-    def order_string(self):
+    def order_string(self) -> str:
         """
         Return the string that defines the monomial order of this algebra.
 
@@ -195,7 +195,7 @@ def order_string(self):
         return self._ordstr
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the generators of this algebra (idempotents and arrows).
 

src/sage/quivers/path_semigroup.py

@@ -468,7 +468,7 @@ def gen(self, i):
         return self.gens()[i]
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the tuple of generators.
 
@@ -487,7 +487,7 @@ def gens(self):
         """
         return self.idempotents() + self.arrows()
 
-    def is_finite(self):
+    def is_finite(self) -> bool:
         """
         This partial semigroup is finite if and only if the underlying
         quiver is acyclic.