sagemathgh-37302: use Parent instead of Algebra in finite_gca

    
This is removing one usage of the auld-class `Algebra` in the modified
file

Also changing a few minor details in doc and code there, and sorting the
imports.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#37302
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

src/sage/algebras/finite_gca.py

@@ -17,18 +17,18 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 from __future__ import annotations
-from sage.combinat.free_module import CombinatorialFreeModule
+
 from sage.categories.algebras import Algebras
-from sage.misc.cachefunc import cached_method
+from sage.combinat.free_module import CombinatorialFreeModule
 from sage.combinat.integer_vector_weighted import WeightedIntegerVectors
-from sage.rings.ring import Algebra
-from sage.misc.functional import is_odd, is_even
-from sage.sets.disjoint_union_enumerated_sets import DisjointUnionEnumeratedSets
-from sage.sets.condition_set import ConditionSet
+from sage.misc.cachefunc import cached_method
+from sage.misc.functional import is_even
 from sage.rings.integer_ring import ZZ
+from sage.sets.condition_set import ConditionSet
+from sage.sets.disjoint_union_enumerated_sets import DisjointUnionEnumeratedSets
 
 
-class FiniteGCAlgebra(CombinatorialFreeModule, Algebra):
+class FiniteGCAlgebra(CombinatorialFreeModule):
     r"""
     Finite dimensional graded commutative algebras.
 
@@ -165,7 +165,7 @@ def __classcall_private__(cls, base, names=None, degrees=None,
                 raise ValueError("You must specify names or degrees")
             else:
                 n = len(degrees)
-            names = tuple('x{}'.format(i) for i in range(n))
+            names = tuple(f'x{i}' for i in range(n))
         elif isinstance(names, str):
             names = tuple(names.split(','))
             n = len(names)
@@ -194,7 +194,6 @@ def __init__(self, base, names, degrees, max_degree,
             sage: TestSuite(A).run()
             sage: A = GradedCommutativeAlgebra(QQ, ('x','y','z','t'), [1,2,3,4], max_degree=10)
             sage: TestSuite(A).run()
-
         """
         from sage.arith.misc import gcd
 
@@ -220,7 +219,7 @@ def __init__(self, base, names, degrees, max_degree,
                                          sorting_key=sorting_key,
                                          category=category)
 
-    def _valid_index(self, w):
+    def _valid_index(self, w) -> bool:
         r"""
         Return whether ``w`` is a valid index; no multiple powers in odd
         degrees.
@@ -234,11 +233,10 @@ def _valid_index(self, w):
             True
             sage: A._valid_index(w2)
             False
-
         """
-        return not any(i > 1 for i, d in zip(w, self._degrees) if is_odd(d))
+        return not any(i > 1 for i, d in zip(w, self._degrees) if d % 2)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.
 
@@ -249,7 +247,6 @@ def _repr_(self):
             "Graded commutative algebra with generators ('x', 'y', 'z') in degrees (1, 2, 3) with maximal degree 8"
             sage: A  # indirect doctest
             Graded commutative algebra with generators ('x', 'y', 'z') in degrees (1, 2, 3) with maximal degree 8
-
         """
         desc = f'Graded commutative algebra with generators {self._names} in '
         desc += f'degrees {self._degrees} with maximal degree {self._max_deg}'
@@ -264,7 +261,6 @@ def ngens(self):
             sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=(4,8,2), max_degree=10)
             sage: A.ngens()
             3
-
         """
         return self.__ngens
 
@@ -327,7 +323,6 @@ def product_on_basis(self, w1, w2):
             x*y^2*z
             sage: A.product_on_basis(w2, w1)
             -x*y^2*z
-
         """
         grading = self._weighted_vectors.grading
         deg_left = grading(w1)
@@ -375,11 +370,10 @@ def degree_on_basis(self, i):
             sage: i = A._weighted_vectors([1,1,0])
             sage: A.degree_on_basis(i)
             6
-
         """
         return self._weighted_vectors.grading(i)
 
-    def _repr_term(self, w):
+    def _repr_term(self, w) -> str:
         r"""
         Return the string representation of basis with index ``w``.
 
@@ -400,7 +394,6 @@ def _repr_term(self, w):
             'x⌣y^2⌣z'
             sage: x*y^2*z  # indirect doctest
             x⌣y^2⌣z
-
         """
         # Trivial case:
         if sum(w) == 0:
@@ -416,7 +409,7 @@ def _repr_term(self, w):
                 terms.append(self._names[i] + f'^{w[i]}')
         return self._mul_symbol.join(terms)
 
-    def _latex_term(self, w):
+    def _latex_term(self, w) -> str:
         r"""
         Return the LaTeX representation of basis with index ``w``.
 
@@ -436,7 +429,6 @@ def _latex_term(self, w):
             'x\\smile y^{2}\\smile z'
             sage: latex(x*y^2*z)  # indirect doctest
             x\smile y^{2}\smile z
-
         """
         # Trivial case:
         if sum(w) == 0:
@@ -463,10 +455,8 @@ def algebra_generators(self):
             sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=(4,8,2), max_degree=10)
             sage: A.algebra_generators()
             Family (x, y, z)
-
         """
         from sage.sets.family import Family
-
         return Family(self.gens())
 
     @cached_method
@@ -483,7 +473,6 @@ def one_basis(self):
             1
             sage: A.one()  # indirect doctest
             1
-
         """
         n = len(self._degrees)
         return self._weighted_vectors([0 for _ in range(n)])
@@ -521,7 +510,6 @@ def gen(self, i):
             y
             sage: A.gen(2)
             z
-
         """
         return self.gens()[i]
 
@@ -534,7 +522,6 @@ def maximal_degree(self):
             sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=(1,2,3), max_degree=8)
             sage: A.maximal_degree()
             8
-
         """
         return self._max_deg