introduce the category of Bounded posets

src/sage/categories/category_with_axiom.py

@@ -1689,6 +1689,7 @@ class ``Sets.Finite``), or in a separate file (typically in a class
                "Commutative", "Cocommutative", "Associative",
                "Inverse", "Unital", "Division", "NoZeroDivisors", "Cellular",
                "AdditiveCommutative", "AdditiveAssociative", "AdditiveInverse", "AdditiveUnital",
+               "Bounded",
                "Extremal", "Trim", "Semidistributive", "CongruenceUniform",
                "Distributive", "Stone",
                "Endset",

src/sage/categories/finite_lattice_posets.py

@@ -9,6 +9,8 @@
 # *****************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
+from sage.categories.posets import Posets
+from sage.misc.cachefunc import cached_method
 
 
 class FiniteLatticePosets(CategoryWithAxiom):
@@ -19,9 +21,11 @@ class FiniteLatticePosets(CategoryWithAxiom):
     EXAMPLES::
 
         sage: FiniteLatticePosets()
-        Category of finite lattice posets
+        Category of finite bounded lattice posets
         sage: FiniteLatticePosets().super_categories()
-        [Category of lattice posets, Category of finite posets]
+        [Category of lattice posets,
+         Category of finite posets,
+         Category of bounded posets]
         sage: FiniteLatticePosets().example()
         NotImplemented
 
@@ -37,6 +41,20 @@ class FiniteLatticePosets(CategoryWithAxiom):
         True
         sage: TestSuite(C).run()
     """
+    @cached_method
+    def extra_super_categories(self):
+        r"""
+        Return a list of the (immediate) super categories of
+        ``self``, as per :meth:`Category.super_categories`.
+
+        EXAMPLES::
+
+            sage: FiniteLatticePosets().super_categories()
+            [Category of lattice posets,
+             Category of finite posets,
+             Category of bounded posets]
+        """
+        return [Posets().Bounded()]
 
     class ParentMethods:
 

src/sage/categories/lattice_posets.py

@@ -201,10 +201,10 @@ class Extremal(CategoryWithAxiom):
         EXAMPLES::
 
             sage: cat = FiniteLatticePosets().Extremal(); cat
-            Category of finite extremal lattice posets
+            Category of finite bounded extremal lattice posets
 
             sage: cat.super_categories()
-            [Category of finite lattice posets,
+            [Category of finite bounded lattice posets,
              Category of extremal lattice posets]
         """
         class ParentMethods:
@@ -226,9 +226,9 @@ class Trim(CategoryWithAxiom):
         EXAMPLES::
 
             sage: cat = FiniteLatticePosets().Trim(); cat
-            Category of finite trim lattice posets
+            Category of finite bounded trim lattice posets
             sage: cat.super_categories()
-            [Category of finite lattice posets,
+            [Category of finite bounded lattice posets,
              Category of trim lattice posets]
         """
         @cached_method
@@ -241,7 +241,7 @@ def extra_super_categories(self):
             EXAMPLES::
 
                 sage: FiniteLatticePosets().Trim().super_categories()
-                [Category of finite lattice posets,
+                [Category of finite bounded lattice posets,
                  Category of trim lattice posets]
             """
             return [LatticePosets().Extremal()]
@@ -265,10 +265,10 @@ class Semidistributive(CategoryWithAxiom):
         EXAMPLES::
 
             sage: cat = FiniteLatticePosets().Semidistributive(); cat
-            Category of finite semidistributive lattice posets
+            Category of finite bounded semidistributive lattice posets
 
             sage: cat.super_categories()
-            [Category of finite lattice posets,
+            [Category of finite bounded lattice posets,
              Category of semidistributive lattice posets]
         """
         class ParentMethods:
@@ -290,9 +290,9 @@ class CongruenceUniform(CategoryWithAxiom):
         EXAMPLES::
 
             sage: cat = FiniteLatticePosets().CongruenceUniform(); cat
-            Category of finite congruence uniform lattice posets
+            Category of finite bounded congruence uniform lattice posets
             sage: cat.super_categories()
-            [Category of finite lattice posets,
+            [Category of finite bounded lattice posets,
              Category of congruence uniform lattice posets]
         """
         @cached_method
@@ -305,7 +305,7 @@ def extra_super_categories(self):
             EXAMPLES::
 
                 sage: FiniteLatticePosets().CongruenceUniform().super_categories()
-                [Category of finite lattice posets,
+                [Category of finite bounded lattice posets,
                  Category of congruence uniform lattice posets]
             """
             return [LatticePosets().Semidistributive()]
@@ -329,10 +329,10 @@ class Distributive(CategoryWithAxiom):
         EXAMPLES::
 
             sage: cat = FiniteLatticePosets().Distributive(); cat
-            Category of finite distributive lattice posets
+            Category of finite bounded distributive lattice posets
 
             sage: cat.super_categories()
-            [Category of finite lattice posets,
+            [Category of finite bounded lattice posets,
              Category of distributive lattice posets]
         """
         @cached_method
@@ -345,7 +345,7 @@ def extra_super_categories(self):
             EXAMPLES::
 
                 sage: FiniteLatticePosets().Distributive().super_categories()
-                [Category of finite lattice posets,
+                [Category of finite bounded lattice posets,
                  Category of distributive lattice posets]
             """
             return [LatticePosets().Trim(),
@@ -371,10 +371,10 @@ class Stone(CategoryWithAxiom):
         EXAMPLES::
 
             sage: cat = FiniteLatticePosets().Stone(); cat
-            Category of finite stone lattice posets
+            Category of finite bounded stone lattice posets
 
             sage: cat.super_categories()
-            [Category of finite lattice posets,
+            [Category of finite bounded lattice posets,
              Category of stone lattice posets]
         """
         @cached_method
@@ -387,7 +387,7 @@ def extra_super_categories(self):
             EXAMPLES::
 
                 sage: FiniteLatticePosets().Stone().super_categories()
-                [Category of finite lattice posets,
+                [Category of finite bounded lattice posets,
                  Category of stone lattice posets]
             """
             return [LatticePosets().Distributive()]

src/sage/categories/posets.py

@@ -12,6 +12,7 @@
 from sage.misc.abstract_method import abstract_method
 from sage.misc.lazy_import import LazyImport
 from sage.categories.category import Category
+from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.sets_cat import Sets
 
 
@@ -719,3 +720,41 @@ class ElementMethods:
         #         sage: x <= y
         #     """
         #     return self.parent().le(self, other)
+
+    class SubcategoryMethods:
+        def Bounded(self):
+            r"""
+            A bounded poset is a poset with a unique maximal element
+            and a unique minimal element.
+
+            EXAMPLES::
+
+                sage: P = posets.DivisorLattice(24)
+                sage: P in Posets().Bounded()
+                True
+            """
+            return self._with_axiom("Bounded")
+
+    class Bounded(CategoryWithAxiom):
+        """
+        The category of bounded posets.
+
+        EXAMPLES::
+
+            sage: cat = Posets().Bounded(); cat
+            Category of bounded posets
+
+            sage: cat.super_categories()
+            [Category of posets]
+        """
+        class ParentMethods:
+            def is_bounded(self):
+                """
+                Return whether ``self`` is a bounded poset.
+
+                EXAMPLES::
+
+                    sage: posets.TamariLattice(4).is_bounded()
+                    True
+                """
+                return True

src/sage/combinat/alternating_sign_matrix.py

@@ -1048,7 +1048,7 @@ 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 bounded lattice posets
     """
 
     def __init__(self, n):

src/sage/combinat/posets/lattices.py

@@ -713,7 +713,7 @@ def LatticePoset(data=None, *args, **options):
 
         sage: L = LatticePoset([[1,2],[3],[3]], facade = True)
         sage: L.category()
-        Category of facade finite enumerated lattice posets
+        Category of facade finite enumerated bounded lattice posets
         sage: parent(L[0])
         Integer Ring
         sage: TestSuite(L).run(skip = ['_test_an_element']) # is_parent_of is not yet implemented