From dcbdf00ffcb2f05292ba3f3d336aa07843b13106 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Jori=20M=C3=A4ntysalo?= <jori.mantysalo@uta.fi>
Date: Thu, 7 Jul 2016 13:17:30 +0300
Subject: [PATCH 1/3] Add certificate to is_relatively_complemented().

---
 src/sage/combinat/posets/lattices.py | 38 +++++++++++++++++++++-------
 1 file changed, 29 insertions(+), 9 deletions(-)

diff --git a/src/sage/combinat/posets/lattices.py b/src/sage/combinat/posets/lattices.py
index 2b98f555758..0c45f246db8 100644
--- a/src/sage/combinat/posets/lattices.py
+++ b/src/sage/combinat/posets/lattices.py
@@ -802,7 +802,7 @@ def is_complemented(self):
         """
         return self._hasse_diagram.is_complemented_lattice()
 
-    def is_relatively_complemented(self):
+    def is_relatively_complemented(self, certificate=False):
         """
         Return ``True`` if the lattice is relatively complemented, and
         ``False`` otherwise.
@@ -810,6 +810,13 @@ def is_relatively_complemented(self):
         A lattice is relatively complemented if every interval of it
         is a complemented lattice.
 
+        INPUT:
+
+        - ``certificate``, a Boolean -- If ``False`` (the default), return
+          only truth value. If ``True``, return either
+          ``(True, None)`` or ``(False, (a, b, c))``, where `b` is the
+          only element that covers `a` and is covered by `c`.
+
         EXAMPLES::
 
             sage: L = LatticePoset({1: [2, 3, 4, 8], 2: [5, 6], 3: [5, 7],
@@ -833,6 +840,11 @@ def is_relatively_complemented(self):
             sage: L.is_relatively_complemented()
             False
 
+        We can also get a non-complemented 3-element interval::
+
+            sage: L.is_relatively_complemented(certificate=True)
+            (False, (1, 6, 11))
+
         TESTS::
 
             sage: [Posets.ChainPoset(i).is_relatively_complemented() for
@@ -863,19 +875,27 @@ def is_relatively_complemented(self):
         H = self._hasse_diagram
         n = H.order()
         if n < 3:
-            return True
+            return (True, None) if certificate else True
 
         # Quick check: the lattice must be atomic and coatomic.
-        if H.out_degree(0) != H.in_degree().count(1):
-            return False
-        if H.in_degree(n-1) != H.out_degree().count(1):
-            return False
+        if not certificate:
+            if H.out_degree(0) != H.in_degree().count(1):
+                return False
+            if H.in_degree(n-1) != H.out_degree().count(1):
+                return False
 
         for e1 in range(n-1):
             C = Counter(flatten([H.neighbors_out(e2) for e2 in H.neighbors_out(e1)]))
-            if any(c == 1 and len(H.closed_interval(e1, e3)) == 3 for e3, c in C.iteritems()):
-                return False
-        return True
+            for e3, c in C.iteritems():
+                if c == 1 and len(H.closed_interval(e1, e3)) == 3:
+                    if not certificate:
+                        return False
+                    e2 = H.neighbors_in(e3)[0]
+                    e1 = H.neighbors_in(e2)[0]
+                    return (False, (self._vertex_to_element(e1),
+                                    self._vertex_to_element(e2),
+                                    self._vertex_to_element(e3)))
+        return (True, None) if certificate else True
 
     def breadth(self, certificate=False):
         r"""

From f8aca83288afbeea6a78ff1db2787b2a2f808234 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Jori=20M=C3=A4ntysalo?= <jori.mantysalo@uta.fi>
Date: Fri, 8 Jul 2016 10:17:36 +0300
Subject: [PATCH 2/3] Restructure input-output blocks.

---
 src/sage/combinat/posets/lattices.py | 13 +++++++++----
 1 file changed, 9 insertions(+), 4 deletions(-)

diff --git a/src/sage/combinat/posets/lattices.py b/src/sage/combinat/posets/lattices.py
index 0c45f246db8..45e0c0409db 100644
--- a/src/sage/combinat/posets/lattices.py
+++ b/src/sage/combinat/posets/lattices.py
@@ -812,10 +812,15 @@ def is_relatively_complemented(self, certificate=False):
 
         INPUT:
 
-        - ``certificate``, a Boolean -- If ``False`` (the default), return
-          only truth value. If ``True``, return either
-          ``(True, None)`` or ``(False, (a, b, c))``, where `b` is the
-          only element that covers `a` and is covered by `c`.
+        - ``certificate`` -- (default: ``False``) Whether to return
+          a certificate if the lattice is not relatively complemented.
+
+        OUTPUT:
+
+        - If ``certificate=True`` return either ``(True, None)`` or
+          ``(False, (a, b, c))``, where `b` is the only element that
+          covers `a` and is covered by `c`.
+        - If ``certificate=False`` return ``True`` or ``False``.
 
         EXAMPLES::
 

From 1213a37113d442434e2dba2ef211f33fd9ec18e3 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Jori=20M=C3=A4ntysalo?= <jori.mantysalo@uta.fi>
Date: Mon, 11 Jul 2016 22:02:17 +0300
Subject: [PATCH 3/3] Output block format.

---
 src/sage/combinat/posets/lattices.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/sage/combinat/posets/lattices.py b/src/sage/combinat/posets/lattices.py
index 45e0c0409db..06429e84083 100644
--- a/src/sage/combinat/posets/lattices.py
+++ b/src/sage/combinat/posets/lattices.py
@@ -819,8 +819,8 @@ def is_relatively_complemented(self, certificate=False):
 
         - If ``certificate=True`` return either ``(True, None)`` or
           ``(False, (a, b, c))``, where `b` is the only element that
-          covers `a` and is covered by `c`.
-        - If ``certificate=False`` return ``True`` or ``False``.
+          covers `a` and is covered by `c`. If ``certificate=False``
+          return ``True`` or ``False``.
 
         EXAMPLES::