Trac #30200: fix the implementation of the "majorization order."

I have used the phrase "majorization order" in the cone catalogue in a
few places. It is however only a preorder, and not a partial order, so
the first order of business is to change that phrase.

There was also one test case that used a majorized_by() function whose
implementation was incorrect. No harm was caused by this, but as a
personal rule I prefer it if a function called majorized_by() actually
implements the majorization order. This commit fixes that as well, by
rearranging the arguments' elements in nonincreasing order, so that
the function truly tests whether or not one argument majorizes the
other.

src/sage/geometry/cone_catalog.py

@@ -239,7 +239,7 @@ def downward_monotonic(ambient_dim=None, lattice=None):
         1
 
     The dual of the downward-monotonic cone is the Schur cone
-    [GS2010]_ that induces the majorization ordering::
+    [GS2010]_ that induces the majorization preordering::
 
         sage: ambient_dim = ZZ.random_element(10)
         sage: K = cones.downward_monotonic(ambient_dim).dual()
@@ -627,8 +627,8 @@ def schur(ambient_dim=None, lattice=None):
     The Schur cone in ``ambient_dim`` dimensions, or living
     in ``lattice``.
 
-    The Schur cone in `n` dimensions induces the majorization ordering
-    on the ambient space. If `\left\{e_{1}, e_{2}, \ldots,
+    The Schur cone in `n` dimensions induces the majorization
+    preordering on the ambient space. If `\left\{e_{1}, e_{2}, \ldots,
     e_{n}\right\}` is the standard basis for the space, then its
     generators are `\left\{e_{i} - e_{i+1}\ |\ 1 \le i \le
     n-1\right\}`. Its dual is the downward monotonic cone.
@@ -702,21 +702,25 @@ def schur(ambient_dim=None, lattice=None):
         sage: cones.schur(0).is_trivial()
         True
 
-    The Schur cone induces the majorization ordering, as in Iusem
-    and Seeger's [IS2005]_ Example 7.3::
+    The Schur cone induces the majorization preordering, as in Iusem
+    and Seeger's [IS2005]_ Example 7.3 or Niezgoda's [Niez1998]_
+    Example 2.2::
 
+        sage: ambient_dim = ZZ.random_element(10)
+        sage: V = VectorSpace(QQ, ambient_dim)
+        sage: rearrange = lambda z: V(sorted(z.list(),reverse=True))
         sage: def majorized_by(x,y):
+        ....:     x = rearrange(x)
+        ....:     y = rearrange(y)
         ....:     return (all(sum(x[0:i]) <= sum(y[0:i])
         ....:                 for i in range(x.degree()-1))
         ....:             and sum(x) == sum(y))
-        sage: ambient_dim = ZZ.random_element(10)
-        sage: V = VectorSpace(QQ, ambient_dim)
         sage: S = cones.schur(ambient_dim)
         sage: majorized_by(V.zero(), S.random_element())
         True
         sage: x = V.random_element()
         sage: y = V.random_element()
-        sage: majorized_by(x,y) == ( (y-x) in S )
+        sage: majorized_by(x,y) == ((rearrange(y)-rearrange(x)) in S)
         True
 
     If a ``lattice`` was given, it is actually used::