gh-39554: creating axioms for lattices
    
as this would be a natural place to put some specific methods from
#37601

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.
    
URL: #39554
Reported by: Frédéric Chapoton
Reviewer(s): Frédéric Chapoton, Travis Scrimshaw

Author: Release Manager

src/sage/categories/category.py

@@ -125,7 +125,8 @@
 _join_cache = WeakValueDictionary()
 
 
-HALL_OF_FAME = ['Coxeter', 'Hopf', 'Weyl', 'Lie', 'Hecke', 'Dedekind']
+HALL_OF_FAME = ['Coxeter', 'Hopf', 'Weyl', 'Lie',
+                'Hecke', 'Dedekind', 'Stone']
 
 
 class Category(UniqueRepresentation, SageObject):

src/sage/categories/category_with_axiom.py

@@ -1658,6 +1658,7 @@ class ``Sets.Finite``), or in a separate file (typically in a class
 
 import importlib
 import re
+
 from sage.misc.cachefunc import cached_method, cached_function
 from sage.misc.lazy_attribute import lazy_class_attribute
 from sage.misc.lazy_import import LazyImport
@@ -1678,21 +1679,23 @@ class ``Sets.Finite``), or in a separate file (typically in a class
                "Differentiable", "Smooth", "Analytic", "AlmostComplex",
                "FinitelyGeneratedAsMagma",
                "WellGenerated",
-               "Facade", "Finite", "Infinite","Enumerated",
+               "Facade", "Finite", "Infinite", "Enumerated",
                "Complete",
                "Nilpotent",
                "FiniteDimensional", "FinitelyPresented", "Connected",
                "FinitelyGeneratedAsLambdaBracketAlgebra",
                "WithBasis",
                "Irreducible",
                "Supercommutative", "Supercocommutative",
-               "Commutative", "Cocommutative", "Associative", "Inverse", "Unital", "Division", "NoZeroDivisors", "Cellular",
+               "Commutative", "Cocommutative", "Associative",
+               "Inverse", "Unital", "Division", "NoZeroDivisors", "Cellular",
                "AdditiveCommutative", "AdditiveAssociative", "AdditiveInverse", "AdditiveUnital",
-               "Distributive",
+               "Extremal", "Trim", "Semidistributive", "CongruenceUniform",
+               "Distributive", "Stone",
                "Endset",
                "Pointed",
-               "Stratified",
-              )
+               "Stratified"
+               )
 
 
 def uncamelcase(s, separator=" "):

src/sage/categories/finite_lattice_posets.py

@@ -2,14 +2,15 @@
 r"""
 Finite lattice posets
 """
-#*****************************************************************************
+# ****************************************************************************
 #  Copyright (C) 2011 Nicolas M. Thiery <nthiery at users.sf.net>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
-#                  http://www.gnu.org/licenses/
-#******************************************************************************
+#                  https://www.gnu.org/licenses/
+# *****************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
+from sage.misc.cachefunc import cached_method
 
 
 class FiniteLatticePosets(CategoryWithAxiom):
@@ -28,7 +29,8 @@ class FiniteLatticePosets(CategoryWithAxiom):
 
     .. SEEALSO::
 
-        :class:`FinitePosets`, :class:`LatticePosets`, :class:`~sage.combinat.posets.lattices.FiniteLatticePoset`
+        :class:`FinitePosets`, :class:`LatticePosets`,
+        :class:`~sage.combinat.posets.lattices.FiniteLatticePoset`
 
     TESTS::
 
@@ -107,7 +109,7 @@ def meet_irreducibles(self):
 
         def meet_irreducibles_poset(self):
             r"""
-            Return the poset of join-irreducible elements of this finite lattice.
+            Return the poset of join-irreducible elements of the lattice.
 
             A *meet-irreducible element* of ``self`` is an element `x`
             that is not maximal and can not be written as the meet of two
@@ -116,7 +118,7 @@ def meet_irreducibles_poset(self):
             EXAMPLES::
 
                 sage: L = LatticePoset({0:[1,2,3],1:[4],2:[4],3:[4]})                   # needs sage.graphs sage.modules
-                sage: L.join_irreducibles_poset()                                       # needs sage.graphs sage.modules
+                sage: L.meet_irreducibles_poset()                                       # needs sage.graphs sage.modules
                 Finite poset containing 3 elements
 
             .. SEEALSO::
@@ -166,10 +168,10 @@ def irreducibles_poset(self):
                 return Poset({self[0]: []})
             return self.subposet(self.join_irreducibles()+self.meet_irreducibles())
 
-        ##########################################################################
+        ######################################################################
         # Lattice morphisms
 
-        def is_lattice_morphism(self, f, codomain):
+        def is_lattice_morphism(self, f, codomain) -> bool:
             r"""
             Return whether ``f`` is a morphism of posets from ``self``
             to ``codomain``.
@@ -234,9 +236,9 @@ def is_lattice_morphism(self, f, codomain):
             # ensure that this is a poset morphism. It actually may
             # be sufficient to check just joins (or just meets).
             from sage.combinat.subset import Subsets
-            for x,y in Subsets(self,2):
-                if f(self.join(x,y)) != codomain.join(f(x), f(y)):
+            for x, y in Subsets(self, 2):
+                if f(self.join(x, y)) != codomain.join(f(x), f(y)):
                     return False
-                if f(self.meet(x,y)) != codomain.meet(f(x), f(y)):
+                if f(self.meet(x, y)) != codomain.meet(f(x), f(y)):
                     return False
             return True

src/sage/categories/finite_posets.py

@@ -1930,26 +1930,33 @@ def order_ideals_lattice(self, as_ideals=True, facade=None):
                 False
             """
             from sage.combinat.posets.lattices import LatticePoset
+            from sage.categories.finite_lattice_posets import FiniteLatticePosets
             if facade is None:
                 facade = self._is_facade
+
             if as_ideals:
                 from sage.misc.call import attrcall
                 from sage.sets.set import Set
                 ideals = [Set(self.order_ideal(antichain))
                           for antichain in self.antichains()]
-                return LatticePoset((ideals, attrcall("issubset")),
-                                    facade=facade)
-            else:
-                from sage.misc.cachefunc import cached_function
-                antichains = [tuple(a) for a in self.antichains()]
+                T = LatticePoset((ideals, attrcall("issubset")),
+                                 facade=facade,
+                                 category=FiniteLatticePosets().Distributive())
+                return T
+
+            from sage.misc.cachefunc import cached_function
+            antichains = [tuple(a) for a in self.antichains()]
+
+            @cached_function
+            def is_above(a, xb):
+                return any(self.is_lequal(xa, xb) for xa in a)
 
-                @cached_function
-                def is_above(a, xb):
-                    return any(self.is_lequal(xa, xb) for xa in a)
+            def compare(a, b):
+                return all(is_above(a, xb) for xb in b)
 
-                def compare(a, b):
-                    return all(is_above(a, xb) for xb in b)
-                return LatticePoset((antichains, compare), facade=facade)
+            T = LatticePoset((antichains, compare), facade=facade,
+                             category=FiniteLatticePosets().Distributive())
+            return T
 
         @abstract_method(optional=True)
         def antichains(self):