BUG: name retention in Index.intersection

pandas/core/indexes/base.py

@@ -2820,7 +2820,7 @@ def intersection(self, other, sort=False):
         """
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
-        other = ensure_index(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
             return self._get_reconciled_name_object(other)

pandas/core/indexes/datetimelike.py

@@ -686,10 +686,15 @@ def intersection(self, other, sort=False):
         """
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
             return self._get_reconciled_name_object(other)
 
+        return self._intersection(other, sort=sort)
+
+    def _intersection(self, other, sort=False):
+
         if len(self) == 0:
             return self.copy()._get_reconciled_name_object(other)
         if len(other) == 0:
@@ -704,10 +709,11 @@ def intersection(self, other, sort=False):
             return result
 
         elif not self._can_fast_intersect(other):
-            result = Index.intersection(self, other, sort=sort)
-            # We need to invalidate the freq because Index.intersection
+            result = Index._intersection(self, other, sort=sort)
+            # We need to invalidate the freq because Index._intersection
             #  uses _shallow_copy on a view of self._data, which will preserve
             #  self.freq if we're not careful.
+            result = self._wrap_setop_result(other, result)
             return result._with_freq(None)._with_freq("infer")
 
         # to make our life easier, "sort" the two ranges

pandas/core/indexes/interval.py

@@ -122,8 +122,13 @@ def setop_check(method):
 
     @wraps(method)
     def wrapped(self, other, sort=False):
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
-        other = ensure_index(other)
+        other, _ = self._convert_can_do_setop(other)
+
+        if op_name == "intersection":
+            if self.equals(other):
+                return self._get_reconciled_name_object(other)
 
         if not isinstance(other, IntervalIndex):
             result = getattr(self.astype(object), op_name)(other)

pandas/core/indexes/multi.py

@@ -3598,10 +3598,15 @@ def intersection(self, other, sort=False):
         """
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
-        other, result_names = self._convert_can_do_setop(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
-            return self.rename(result_names)
+            return self._get_reconciled_name_object(other)
+
+        return self._intersection(other, sort=sort)
+
+    def _intersection(self, other, sort=False):
+        other, result_names = self._convert_can_do_setop(other)
 
         if not is_object_dtype(other.dtype):
             # The intersection is empty
@@ -3717,11 +3722,12 @@ def _convert_can_do_setop(self, other):
                     levels=[[]] * self.nlevels,
                     codes=[[]] * self.nlevels,
                     verify_integrity=False,
+                    names=self.names,
                 )
             else:
                 msg = "other must be a MultiIndex or a list of tuples"
                 try:
-                    other = MultiIndex.from_tuples(other)
+                    other = MultiIndex.from_tuples(other, names=self.names)
                 except TypeError as err:
                     raise TypeError(msg) from err
         else:

pandas/core/indexes/period.py

@@ -639,15 +639,19 @@ def _setop(self, other, sort, opname: str):
     def intersection(self, other, sort=False):
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
-        other = ensure_index(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
             return self._get_reconciled_name_object(other)
 
-        elif is_object_dtype(other.dtype):
+        return self._intersection(other, sort=sort)
+
+    def _intersection(self, other, sort=False):
+
+        if is_object_dtype(other.dtype):
             return self.astype("O").intersection(other, sort=sort)
 
-        elif not is_dtype_equal(self.dtype, other.dtype):
+        elif not self._is_comparable_dtype(other.dtype):
             # We can infer that the intersection is empty.
             # assert_can_do_setop ensures that this is not just a mismatched freq
             this = self[:0].astype("O")

pandas/core/indexes/range.py

@@ -504,12 +504,19 @@ def intersection(self, other, sort=False):
         intersection : Index
         """
         self._validate_sort_keyword(sort)
+        self._assert_can_do_setop(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
             return self._get_reconciled_name_object(other)
 
+        return self._intersection(other, sort=sort)
+
+    def _intersection(self, other, sort=False):
+
         if not isinstance(other, RangeIndex):
-            return super().intersection(other, sort=sort)
+            result = super()._intersection(other, sort=sort)
+            return self._wrap_setop_result(other, result)
 
         if not len(self) or not len(other):
             return self._simple_new(_empty_range)

pandas/tests/indexes/datetimes/test_setops.py

@@ -471,10 +471,11 @@ def test_intersection_bug(self):
 
     def test_intersection_list(self):
         # GH#35876
+        # values is not an Index -> no name -> retain "a"
         values = [pd.Timestamp("2020-01-01"), pd.Timestamp("2020-02-01")]
         idx = DatetimeIndex(values, name="a")
         res = idx.intersection(values)
-        tm.assert_index_equal(res, idx.rename(None))
+        tm.assert_index_equal(res, idx)
 
     def test_month_range_union_tz_pytz(self, sort):
         from pytz import timezone

pandas/tests/indexes/test_setops.py

@@ -98,13 +98,20 @@ def test_compatible_inconsistent_pairs(idx_fact1, idx_fact2):
         ("Period[D]", "float64", "object"),
     ],
 )
-def test_union_dtypes(left, right, expected):
+@pytest.mark.parametrize("names", [("foo", "foo", "foo"), ("foo", "bar", None)])
+def test_union_dtypes(left, right, expected, names):
     left = pandas_dtype(left)
     right = pandas_dtype(right)
-    a = pd.Index([], dtype=left)
-    b = pd.Index([], dtype=right)
-    result = a.union(b).dtype
-    assert result == expected
+    a = pd.Index([], dtype=left, name=names[0])
+    b = pd.Index([], dtype=right, name=names[1])
+    result = a.union(b)
+    assert result.dtype == expected
+    assert result.name == names[2]
+
+    # Testing name retention
+    # TODO: pin down desired dtype; do we want it to be commutative?
+    result = a.intersection(b)
+    assert result.name == names[2]
 
 
 def test_dunder_inplace_setops_deprecated(index):
@@ -378,6 +385,25 @@ def test_intersect_unequal(self, index, fname, sname, expected_name):
         expected = index[1:].set_names(expected_name).sort_values()
         tm.assert_index_equal(intersect, expected)
 
+    def test_intersection_name_retention_with_nameless(self, index):
+        if isinstance(index, MultiIndex):
+            index = index.rename(list(range(index.nlevels)))
+        else:
+            index = index.rename("foo")
+
+        other = np.asarray(index)
+
+        result = index.intersection(other)
+        assert result.name == index.name
+
+        # empty other, same dtype
+        result = index.intersection(other[:0])
+        assert result.name == index.name
+
+        # empty `self`
+        result = index[:0].intersection(other)
+        assert result.name == index.name
+
     def test_difference_preserves_type_empty(self, index, sort):
         # GH#20040
         # If taking difference of a set and itself, it