diff --git a/graphblas/core/operator/monoid.py b/graphblas/core/operator/monoid.py
index bcf06d323..e6d2ce07e 100644
--- a/graphblas/core/operator/monoid.py
+++ b/graphblas/core/operator/monoid.py
@@ -32,11 +32,22 @@
 def _is_builtin_binaryop(binaryop):
     """True if ``binaryop`` wraps built-in GraphBLAS operators rather than a UDF.
 
-    Built-ins are the only binary ops constructed with neither a Python func nor
-    a numba func. That covers ``binary.numpy.float_power`` too, which is
-    assembled out of ``binary.pow``'s typed ops.
+    Most built-ins are constructed with neither a Python func nor a numba func.
+    That covers ``binary.numpy.float_power`` too, which is assembled out of
+    ``binary.pow``'s typed ops. A few built-ins (``any``, ``first``, ``pair``,
+    ``second``) carry a Python ``orig_func`` as their UDT fallback, so the func
+    test alone misses them; their typed ops are still ``TypedBuiltinBinaryOp``,
+    which is the authoritative signal. A UDF's typed ops are never
+    ``TypedBuiltinBinaryOp``. Under ``mapnumpy=True`` the mapped numpy binary
+    ops hold the built-in's typed ops and no funcs, so both tests agree they
+    are built-in; the numpy monoids themselves are mapped without passing
+    through ``Monoid._build``.
     """
-    return binaryop.orig_func is None and binaryop._numba_func is None
+    return (
+        binaryop.orig_func is None
+        and binaryop._numba_func is None
+        or any(type(typed) is TypedBuiltinBinaryOp for typed in binaryop._typed_ops.values())
+    )
 
 
 def _scalar_identity(monoid_name, scalar_dtype):
@@ -53,7 +64,9 @@ def _scalar_identity(monoid_name, scalar_dtype):
         return scalar_dtype.type(1)
     if monoid_name not in {"min", "max"}:
         return None
-    if np.issubdtype(scalar_dtype, np.integer):
+    # Not ``np.issubdtype(scalar_dtype, np.integer)``: numpy files timedelta64
+    # under ``np.integer``, and ``np.iinfo`` rejects it.
+    if scalar_dtype.kind in "iu":
         info = np.iinfo(scalar_dtype)
         return info.max if monoid_name == "min" else info.min
     if np.issubdtype(scalar_dtype, np.floating):
@@ -340,9 +353,18 @@ def _compile_udt(self, dtype, dtype2):
         from ..scalar import Scalar
 
         ret_type = binaryop.return_type
+        if ret_type != dtype:
+            # Checked here because GrB_Monoid_new_UDT reports it as a DomainMismatch
+            # with no message.
+            raise TypeError(
+                f"monoid.{self.name} does not work with {dtype}: a monoid's binary operator "
+                f"must return its input type, {dtype}, not {ret_type}"
+            )
         identity_val = self._identity
-        if identity_val is None and self.name in _BUILTIN_UDT_MONOIDS:
-            # Auto-generate the identity for built-in monoids on UDTs.
+        if identity_val is None and self._binaryop is None and self.name in _BUILTIN_UDT_MONOIDS:
+            # Auto-generate the identity for built-in monoids on UDTs. Only built-in
+            # monoids have no ``_binaryop``: a user monoid named like one (after its
+            # op's function, or by ``name=``) must not get that identity.
             identity_val = _udt_identity(self.name, ret_type)
         if identity_val is None:
             raise KeyError(
@@ -439,7 +461,12 @@ def register_new(cls, name, binaryop, identity, *, is_idempotent=False, lazy=Fal
         if lazy:
             module._delayed[funcname] = (
                 cls.register_new,
-                {"name": name, "binaryop": binaryop, "identity": identity},
+                {
+                    "name": name,
+                    "binaryop": binaryop,
+                    "identity": identity,
+                    "is_idempotent": is_idempotent,
+                },
             )
         elif type(binaryop) is ParameterizedBinaryOp:
             monoid = ParameterizedMonoid(name, binaryop, identity, is_idempotent=is_idempotent)
diff --git a/graphblas/tests/test_op.py b/graphblas/tests/test_op.py
index c76c895df..594b196a6 100644
--- a/graphblas/tests/test_op.py
+++ b/graphblas/tests/test_op.py
@@ -440,6 +440,24 @@ def test_monoid_rejects_builtin_binaryop():
     assert not hasattr(gb.monoid, "_bad_builtin_monoid")
 
 
+def test_monoid_rejects_builtin_binaryop_with_udt_fallback():
+    """binary.any, first, pair, and second are built-ins that carry a Python
+    func as their UDT fallback, so a func-based probe alone misses them. They
+    must be rejected like the other built-ins: their typed ops have no _monoid
+    slot, and a failed registration used to leave binary.<name>._monoid
+    pointing at the half-built monoid.
+    """
+    for binop in [binary.any, binary.first, binary.pair, binary.second]:
+        with pytest.raises(TypeError, match="must be a user-defined BinaryOp"):
+            Monoid.register_anonymous(binop, 0)
+    # State is untouched: any's built-in monoid association survives, and the
+    # ops without a built-in monoid still have none.
+    assert binary.any.monoid is monoid.any
+    assert binary.first.monoid is None
+    assert binary.pair.monoid is None
+    assert binary.second.monoid is None
+
+
 @pytest.mark.skipif("not supports_udfs")
 @pytest.mark.slow
 def test_semiring_parameterized():
@@ -1188,6 +1206,9 @@ def test_lazy_op():
     Monoid.register_new("lazy2", binary.lazy, 0, lazy=True)
     assert isinstance(op.lazy2, Monoid)
     assert isinstance(monoid.lazy2, Monoid)
+    BinaryOp.register_new("lazy_idem", _isidem_pick_first, lazy=True)
+    Monoid.register_new("lazy_idem", "lazy_idem", 0, is_idempotent=True, lazy=True)
+    assert monoid.lazy_idem.is_idempotent
     Semiring.register_new("lazy", "lazy", "lazy", lazy=True)
     assert isinstance(semiring.lazy, Semiring)
     Semiring.register_new("lazy_lazy", monoid.lazy, binary.lazy, lazy=True)
@@ -1981,6 +2002,27 @@ def _add_udt(x, y):
     assert result.isequal(expected)
 
 
+@pytest.mark.skipif("not supports_udfs")
+@pytest.mark.slow
+def test_udt_monoid_op_must_return_its_input_type(record_udt):
+    """A UDT monoid whose op returns another type fails up front, with a message.
+
+    It used to reach ``GrB_Monoid_new_UDT``, which raised DomainMismatch with an
+    empty message, also from ``udt in monoid``.
+    """
+
+    def _sum_a(x, y):
+        return x["a"] + y["a"]  # pragma: no cover (numba)
+
+    sum_a = BinaryOp.register_anonymous(_sum_a, "test_sum_a_mon", is_udt=True)
+    sum_a_monoid = Monoid.register_anonymous(sum_a, 0)
+    # Not "_BuiltinOpsRec": registering this record dtype again under another name, as
+    # test_udt_op_pickle does, renames the one DataType
+    with pytest.raises(TypeError, match=f"must return its input type, {record_udt}, not INT64"):
+        sum_a_monoid[record_udt]
+    assert record_udt not in sum_a_monoid
+
+
 @pytest.mark.skipif("not supports_udfs")
 @pytest.mark.slow
 def test_udt_tuple_return_semiring(record_udt):
@@ -2456,6 +2498,56 @@ def test_udt_auto_monoid():
     assert any_res in ((7, 8.0), (11, 12.0))
 
 
+@pytest.mark.skipif("not supports_udfs")
+@pytest.mark.slow
+def test_udt_auto_monoid_time_fields():
+    """Monoid min and max generate no identity for a datetime64 or timedelta64 field.
+
+    numpy files timedelta64 under ``np.integer``, so a timedelta64 field used to
+    reach ``np.iinfo`` and raise ``ValueError: Invalid integer data type 'm'``,
+    from ``udt in agg.min`` too, where a datetime64 field gets a KeyError.
+    """
+    for code in ["M8[s]", "m8[s]"]:
+        udt = dtypes.register_anonymous(np.dtype([("tf_n", np.int64), ("tf_t", code)]))
+        for mon in [monoid.min, monoid.max]:
+            with pytest.raises(KeyError, match="field 'tf_t' has unsupported dtype"):
+                mon[udt]
+            assert udt not in mon
+        assert udt not in agg.min
+        assert udt not in agg.max
+
+
+@pytest.mark.skipif("not supports_udfs")
+@pytest.mark.slow
+def test_udt_monoid_named_like_builtin_gets_no_identity():
+    """Only the built-in monoids generate a UDT identity, not a user monoid of the same name.
+
+    A user monoid named plus, times, min or max (after its op's function, or by
+    ``name=``) used to get that built-in's generated identity. The op here takes
+    the field-wise max, so the generated ``min`` identity, INT64_MAX in every
+    field, absorbed every value: the reduce below gave that instead of (5, 2).
+    """
+    udt = dtypes.register_anonymous(np.dtype([("ni_a", np.int64), ("ni_b", np.int64)]))
+    v = Vector(udt, size=2)
+    v[0] = (1, 2)
+    v[1] = (5, -3)
+
+    def _fieldwise_max(x, y):
+        return (max(x["ni_a"], y["ni_a"]), max(x["ni_b"], y["ni_b"]))  # pragma: no cover (numba)
+
+    fieldwise_max = BinaryOp.register_anonymous(_fieldwise_max, "min", is_udt=True)
+    named_by_op = Monoid.register_anonymous(fieldwise_max, None)
+    assert named_by_op.name == "min"
+    for mon in [named_by_op] + [
+        Monoid.register_anonymous(fieldwise_max, None, name) for name in ["plus", "times", "max"]
+    ]:
+        with pytest.raises(KeyError, match="no identity value"):
+            mon[udt]
+        assert udt not in mon
+    lowest = np.iinfo(np.int64).min
+    assert v.reduce(Monoid.register_anonymous(fieldwise_max, (lowest, lowest))).new() == (5, 2)
+
+
 @pytest.mark.skipif("not supports_udfs")
 @pytest.mark.parametrize("shape", [(4,), (3, 2)])
 def test_udt_array_wrapper_stays_within_element(shape):
diff --git a/graphblas/tests/test_operator_types.py b/graphblas/tests/test_operator_types.py
index 8c250bc25..9192b37ae 100644
--- a/graphblas/tests/test_operator_types.py
+++ b/graphblas/tests/test_operator_types.py
@@ -185,6 +185,7 @@
     "plus_copysign",
     "lazy",
     "lazy2",
+    "lazy_idem",
     "lazy_lazy",
     # UDFs created during tests
     "is_positive",