diff --git a/src/sage/categories/category_with_axiom.py b/src/sage/categories/category_with_axiom.py
index 6d7fa75fbf6..2c76e30c529 100644
--- a/src/sage/categories/category_with_axiom.py
+++ b/src/sage/categories/category_with_axiom.py
@@ -1691,7 +1691,7 @@ class ``Sets.Finite``), or in a separate file (typically in a class
                "Inverse", "Unital", "Division", "NoZeroDivisors", "Cellular",
                "AdditiveCommutative", "AdditiveAssociative", "AdditiveInverse", "AdditiveUnital",
                "Extremal", "Trim", "Semidistributive", "CongruenceUniform",
-               "Distributive", "Stone",
+               "Graded", "Distributive", "Stone",
                "Endset",
                "Pointed",
                "Stratified"
diff --git a/src/sage/categories/finite_complex_reflection_groups.py b/src/sage/categories/finite_complex_reflection_groups.py
index 7bb9fda5a81..dad8ffa941f 100644
--- a/src/sage/categories/finite_complex_reflection_groups.py
+++ b/src/sage/categories/finite_complex_reflection_groups.py
@@ -17,6 +17,7 @@
 from sage.misc.cachefunc import cached_method
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.coxeter_groups import CoxeterGroups
+from sage.categories.finite_lattice_posets import FiniteLatticePosets
 
 
 class FiniteComplexReflectionGroups(CategoryWithAxiom):
@@ -866,7 +867,7 @@ def absolute_order_ideal(self, gens=None,
                     14
 
                     sage: W = CoxeterGroup(['A', 2])                                    # needs sage.combinat sage.groups
-                    sage: for (w, l) in W.absolute_order_ideal(return_lengths=True):    # needs sage.combinat sage.groups
+                    sage: for w, l in W.absolute_order_ideal(return_lengths=True):    # needs sage.combinat sage.groups
                     ....:     print(w.reduced_word(), l)
                     [1, 2] 2
                     [1, 2, 1] 1
@@ -967,7 +968,7 @@ def noncrossing_partition_lattice(self, c=None, L=None,
                     L = [(pi, pi.reflection_length()) for pi in L]
                 rels = []
                 ref_lens = dict(L)
-                for (pi, l) in L:
+                for pi, l in L:
                     for t in R:
                         tau = pi * t
                         if tau in ref_lens and l + 1 == ref_lens[tau]:
@@ -975,7 +976,8 @@ def noncrossing_partition_lattice(self, c=None, L=None,
 
                 P = Poset(([], rels), cover_relations=True, facade=True)
                 if P.is_lattice():
-                    P = LatticePoset(P)
+                    P = LatticePoset(P,
+                                     category=FiniteLatticePosets().Graded())
                 return P
 
             def generalized_noncrossing_partitions(self, m, c=None, positive=False):
diff --git a/src/sage/categories/finite_coxeter_groups.py b/src/sage/categories/finite_coxeter_groups.py
index 9abc05fde03..654a21604df 100644
--- a/src/sage/categories/finite_coxeter_groups.py
+++ b/src/sage/categories/finite_coxeter_groups.py
@@ -17,6 +17,7 @@
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.coxeter_groups import CoxeterGroups
+from sage.categories.finite_lattice_posets import FiniteLatticePosets
 
 
 class FiniteCoxeterGroups(CategoryWithAxiom):
@@ -483,8 +484,9 @@ def weak_poset(self, side='right', facade=False):
                 return Poset((self, covers), cover_relations=True,
                              facade=facade)
             covers = tuple([u, v] for u in self for v in u.upper_covers(side=side))
+            cat = FiniteLatticePosets().Graded()
             return LatticePoset((self, covers), cover_relations=True,
-                                facade=facade)
+                                facade=facade, category=cat)
 
         weak_lattice = weak_poset
 
diff --git a/src/sage/categories/lattice_posets.py b/src/sage/categories/lattice_posets.py
index 6f094efdc3b..a1cffd38af1 100644
--- a/src/sage/categories/lattice_posets.py
+++ b/src/sage/categories/lattice_posets.py
@@ -91,6 +91,18 @@ def join(self, x, y):
             """
 
     class SubcategoryMethods:
+        def Graded(self):
+            r"""
+            A lattice is graded if all maximal chains have the same length.
+
+            EXAMPLES::
+
+                sage: P = posets.DivisorLattice(24)
+                sage: P in FiniteLatticePosets().Graded()
+                True
+            """
+            return self._with_axiom("Graded")
+
         def Stone(self):
             r"""
             A Stone lattice `(L, \vee, \wedge)` is a pseudo-complemented
@@ -115,6 +127,8 @@ def Distributive(self):
             From duality in lattices, it follows that then also join
             distributes over meet.
 
+            A distributive lattice is always graded.
+
             See :wikipedia:`Distributive lattice`.
 
             EXAMPLES::
@@ -247,6 +261,18 @@ def extra_super_categories(self):
             """
             return [LatticePosets().Extremal()]
 
+        class SubcategoryMethods:
+            def Graded(self):
+                r"""
+                A trim and graded lattice is distributive.
+
+                EXAMPLES::
+
+                    sage: FiniteLatticePosets().Trim().Graded()
+                    Category of finite distributive lattice posets
+                """
+                return self._with_axiom("Distributive")
+
         class ParentMethods:
             def is_trim(self):
                 """
@@ -350,7 +376,8 @@ def extra_super_categories(self):
                  Category of distributive lattice posets]
             """
             return [LatticePosets().Trim(),
-                    LatticePosets().CongruenceUniform()]
+                    LatticePosets().CongruenceUniform(),
+                    LatticePosets().Graded()]
 
         class ParentMethods:
             def is_distributive(self):
@@ -404,3 +431,28 @@ def is_stone(self):
                     True
                 """
                 return True
+
+    class Graded(CategoryWithAxiom):
+        """
+        The category of graded lattices.
+
+        EXAMPLES::
+
+            sage: cat = FiniteLatticePosets().Graded(); cat
+            Category of finite graded lattice posets
+
+            sage: cat.super_categories()
+            [Category of finite lattice posets,
+             Category of graded lattice posets]
+        """
+        class ParentMethods:
+            def is_graded(self):
+                """
+                Return whether ``self`` is a graded lattice.
+
+                EXAMPLES::
+
+                    sage: posets.DivisorLattice(12).is_graded()
+                    True
+                """
+                return True
diff --git a/src/sage/combinat/alternating_sign_matrix.py b/src/sage/combinat/alternating_sign_matrix.py
index 20b20056cc1..925ff63839e 100644
--- a/src/sage/combinat/alternating_sign_matrix.py
+++ b/src/sage/combinat/alternating_sign_matrix.py
@@ -41,6 +41,7 @@
 from sage.structure.element import Element
 from sage.structure.richcmp import richcmp
 from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
+from sage.categories.finite_lattice_posets import FiniteLatticePosets
 from sage.matrix.matrix_space import MatrixSpace
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import zero_vector
@@ -60,8 +61,8 @@
 
 def _inplace_height_function_gyration(hf):
     k = hf.nrows() - 1
-    for i in range(1,k):
-        for j in range(1,k):
+    for i in range(1, k):
+        for j in range(1, k):
             if (i+j) % 2 == 0 \
                     and hf[i-1,j] == hf[i+1,j] == hf[i,j+1] == hf[i,j-1]:
                 if hf[i,j] < hf[i+1,j]:
@@ -800,11 +801,11 @@ def ASM_compatible_smaller(self):
             N = len(output)
             for c in range(N):
                 d = copy.copy(output[c])
-                output[c][sign[b][0],sign[b][1]] = -output[c][sign[b][0], sign[b][1]]+1
-                d[sign[b][0],sign[b][1]] = -d[sign[b][0], sign[b][1]]-3
+                output[c][sign[b][0], sign[b][1]] = -output[c][sign[b][0], sign[b][1]]+1
+                d[sign[b][0], sign[b][1]] = -d[sign[b][0], sign[b][1]]-3
                 output.append(d)
         for k in range(len(output)):
-            output[k] = M.from_height_function((output[k]-matrix.ones(n,n))/2)
+            output[k] = M.from_height_function((output[k]-matrix.ones(n, n))/2)
         return output
 
     @combinatorial_map(name='to Dyck word')
@@ -1043,10 +1044,10 @@ class AlternatingSignMatrices(UniqueRepresentation, Parent):
         sage: L
         Finite lattice containing 7 elements
         sage: L.category()
-        Category of facade finite enumerated lattice posets
+        Category of facade finite enumerated distributive lattice posets
     """
 
-    def __init__(self, n):
+    def __init__(self, n) -> None:
         r"""
         Initialize ``self``.
 
@@ -1059,7 +1060,7 @@ def __init__(self, n):
         self._matrix_space = MatrixSpace(ZZ, n)
         Parent.__init__(self, category=FiniteEnumeratedSets())
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -1084,7 +1085,7 @@ def _repr_option(self, key):
         """
         return self._matrix_space._repr_option(key)
 
-    def __contains__(self, asm):
+    def __contains__(self, asm) -> bool:
         """
         Check if ``asm`` is in ``self``.
 
@@ -1559,8 +1560,9 @@ def lattice(self):
             sage: L
             Finite lattice containing 7 elements
         """
+        cat = FiniteLatticePosets().Distributive()
         return LatticePoset(self._lattice_initializer(), cover_relations=True,
-                            check=False)
+                            check=False, category=cat)
 
     @cached_method
     def gyration_orbits(self):
diff --git a/src/sage/combinat/posets/bubble_shuffle.py b/src/sage/combinat/posets/bubble_shuffle.py
index 92dd66c72fa..dbe57eeac7f 100644
--- a/src/sage/combinat/posets/bubble_shuffle.py
+++ b/src/sage/combinat/posets/bubble_shuffle.py
@@ -201,7 +201,8 @@ def ShufflePoset(m, n) -> LatticePoset:
     dg = DiGraph([(x, y) for x in bubbles
                   for y in bubble_coverings(m, n, x, transpose=False)])
     # here we just have the cover relations
-    return LatticePoset(dg, cover_relations=True)
+    cat = FiniteLatticePosets().Graded()
+    return LatticePoset(dg, cover_relations=True, check=False, category=cat)
 
 
 def noncrossing_bipartite_complex(m, n):
diff --git a/src/sage/combinat/posets/lattices.py b/src/sage/combinat/posets/lattices.py
index e06fb09627f..d891b15e9c5 100644
--- a/src/sage/combinat/posets/lattices.py
+++ b/src/sage/combinat/posets/lattices.py
@@ -156,7 +156,7 @@
 from sage.combinat.posets.hasse_diagram import LatticeError
 
 
-####################################################################################
+############################################################################
 
 def MeetSemilattice(data=None, *args, **options):
     r"""
@@ -481,7 +481,7 @@ def pseudocomplement(self, element):
             return None
         return self._vertex_to_element(e)
 
-####################################################################################
+############################################################################
 
 
 def JoinSemilattice(data=None, *args, **options):
diff --git a/src/sage/combinat/posets/poset_examples.py b/src/sage/combinat/posets/poset_examples.py
index d8bb880d360..d9b77e60ea6 100644
--- a/src/sage/combinat/posets/poset_examples.py
+++ b/src/sage/combinat/posets/poset_examples.py
@@ -440,7 +440,7 @@ def DiamondPoset(n, facade=None):
         c[0] = list(range(1, n - 1))
         c[n - 1] = []
         D = DiGraph({v: c[v] for v in range(n)}, format='dict_of_lists')
-        cat = FiniteLatticePosets()
+        cat = FiniteLatticePosets().Graded()
         if n <= 4:
             cat = cat.Stone()
         return FiniteLatticePoset(hasse_diagram=D, category=cat,
@@ -1182,14 +1182,14 @@ def SymmetricGroupWeakOrderPoset(n, labels='permutations', side='right'):
         EXAMPLES::
 
             sage: posets.SymmetricGroupWeakOrderPoset(4)
-            Finite poset containing 24 elements
+            Finite lattice containing 24 elements
         """
         if n < 10 and labels == "permutations":
-            element_labels = dict([[s, "".join(map(str, s))]
-                                   for s in Permutations(n)])
+            element_labels = {s: "".join(map(str, s))
+                              for s in Permutations(n)}
         if n < 10 and labels == "reduced_words":
-            element_labels = dict([[s, "".join(map(str, s.reduced_word_lexmin()))]
-                                   for s in Permutations(n)])
+            element_labels = {s: "".join(map(str, s.reduced_word_lexmin()))
+                              for s in Permutations(n)}
         if side == "left":
 
             def weak_covers(s):
@@ -1200,6 +1200,7 @@ def weak_covers(s):
                 return [v for v in s.bruhat_succ() if
                         s.length() + (s.inverse().right_action_product(v)).length() == v.length()]
         else:
+
             def weak_covers(s):
                 r"""
                 Nested function for computing the covers of elements in the
@@ -1207,8 +1208,11 @@ def weak_covers(s):
                 """
                 return [v for v in s.bruhat_succ() if
                         s.length() + (s.inverse().left_action_product(v)).length() == v.length()]
-        return Poset(dict([[s, weak_covers(s)] for s in Permutations(n)]),
-                     element_labels)
+        return LatticePoset(
+            {s: weak_covers(s) for s in Permutations(n)},
+            element_labels, check=False,
+            category=FiniteLatticePosets().Graded().Semidistributive()
+        )
 
     @staticmethod
     def TetrahedralPoset(n, *colors, **labels):
@@ -2125,7 +2129,7 @@ def _random_distributive_lattice(n):
     return D
 
 
-def _random_stone_lattice(n):
+def _random_stone_lattice(n) -> DiGraph:
     """
     Return a random Stone lattice on `n` elements.