Trac #17226: LatticePoset: add Frattini sublattice

Add a function that computes the Frattini sublattice, i.e. intersection
of all proper sublattices of a lattice. AFAIK there is no known good
algorithm for this. Maybe making even a slow function will make it
easier to search for one.

URL: http://trac.sagemath.org/17226
Reported by: jmantysalo
Ticket author(s): Jori Mäntysalo
Reviewer(s): Travis Scrimshaw

src/sage/combinat/posets/hasse_diagram.py

@@ -1636,6 +1636,24 @@ def sublattice(elms, e):
 
         return result
 
+    def frattini_sublattice(self):
+        """
+        Return the list of elements of the Frattini sublattice of the lattice.
+
+        EXAMPLES::
+
+            sage: H = Posets.PentagonPoset()._hasse_diagram
+            sage: H.frattini_sublattice()
+            [0, 4]
+        """
+        # Just a direct computation, no optimization at all.
+        n = self.cardinality()
+        if n == 0 or n == 2: return []
+        if n == 1: return [0]
+        max_sublats = self.maximal_sublattices()
+        return [e for e in range(self.cardinality()) if
+                all(e in ms for ms in max_sublats)]
+
 from sage.misc.rest_index_of_methods import gen_rest_table_index
 import sys
 __doc__ = __doc__.format(INDEX_OF_FUNCTIONS=gen_rest_table_index(HasseDiagram))

src/sage/combinat/posets/lattices.py

@@ -24,6 +24,8 @@
     :delim: |
 
     :meth:`~FiniteLatticePoset.complements` | Return the list of complements of an element, or the dictionary of complements for all elements.
+    :meth:`~FiniteLatticePoset.maximal_sublattices` | Return maximal sublattices of the lattice.
+    :meth:`~FiniteLatticePoset.frattini_sublattice` | Return the intersection of maximal sublattices.
     :meth:`~FiniteLatticePoset.is_atomic` | Return ``True`` if the lattice is atomic.
     :meth:`~FiniteLatticePoset.is_complemented` | Return ``True`` if the lattice is complemented.
     :meth:`~FiniteLatticePoset.is_distributive` | Return ``True`` if the lattice is distributive.
@@ -947,6 +949,31 @@ def maximal_sublattices(self):
             return [self.sublattice([self.bottom()]), self.sublattice([self.top()])]
         return [self.sublattice([self[x] for x in d]) for d in self._hasse_diagram.maximal_sublattices()]
 
+    def frattini_sublattice(self):
+        r"""
+        Return the Frattini sublattice of the lattice.
+
+        The Frattini sublattice `\Phi(L)` is the intersection of all
+        proper maximal sublattices of `L`. It is also the set of
+        "non-generators" - if the sublattice generated by set `S` of
+        elements is whole lattice, then also `S \setminus \Phi(L)`
+        generates whole lattice.
+
+        EXAMPLES::
+
+            sage: L = LatticePoset(( [], [[1,2],[1,17],[1,8],[2,3],[2,22],
+            ....:                         [2,5],[2,7],[17,22],[17,13],[8,7],
+            ....:                         [8,13],[3,16],[3,9],[22,16],[22,18],
+            ....:                         [22,10],[5,18],[5,14],[7,9],[7,14],
+            ....:                         [7,10],[13,10],[16,6],[16,19],[9,19],
+            ....:                         [18,6],[18,33],[14,33],[10,19],
+            ....:                         [10,33],[6,4],[19,4],[33,4]] ))
+            sage: sorted(L.frattini_sublattice().list())
+            [1, 2, 4, 10, 19, 22, 33]
+        """
+        return LatticePoset(self.subposet([self[x] for x in
+                self._hasse_diagram.frattini_sublattice()]))
+
 ############################################################################
 
 FiniteMeetSemilattice._dual_class = FiniteJoinSemilattice