introduce the category of Bounded posets

src/sage/categories/category_with_axiom.py

@@ -1678,6 +1678,7 @@ class ``Sets.Finite``), or in a separate file (typically in a class
                "Differentiable", "Smooth", "Analytic", "AlmostComplex",
                "FinitelyGeneratedAsMagma",
                "WellGenerated",
+               "Bounded",
                "Facade", "Finite", "Infinite", "Enumerated",
                "Complete",
                "Nilpotent",

src/sage/categories/finite_lattice_posets.py

@@ -9,6 +9,8 @@
 # *****************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
+from sage.categories.lattice_posets import LatticePosets
+from sage.misc.cachefunc import cached_method
 
 
 class FiniteLatticePosets(CategoryWithAxiom):
@@ -21,7 +23,9 @@ class FiniteLatticePosets(CategoryWithAxiom):
         sage: FiniteLatticePosets()
         Category of finite 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 +41,19 @@ class FiniteLatticePosets(CategoryWithAxiom):
         True
         sage: TestSuite(C).run()
     """
+    @cached_method
+    def extra_super_categories(self):
+        r"""
+        Implement the fact that a finite lattice is a bounded poset.
+
+        EXAMPLES::
+
+            sage: FiniteLatticePosets().super_categories()
+            [Category of lattice posets,
+             Category of finite posets,
+             Category of bounded posets]
+        """
+        return [LatticePosets().Bounded()]
 
     class ParentMethods:
 

src/sage/categories/lattice_posets.py

@@ -90,6 +90,7 @@ def join(self, x, y):
             """
 
     class SubcategoryMethods:
+
         def Stone(self):
             r"""
             A Stone lattice `(L, \vee, \wedge)` is a pseudo-complemented

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