Add total order to MonomialIndex

1 files changed, 19 insertions, 3 deletions

diff --git a/polymatrix/polymatrix/typing.py b/polymatrix/polymatrix/typing.py
index ce33e67..cebf636 100644
--- a/polymatrix/polymatrix/typing.py
+++ b/polymatrix/polymatrix/typing.py
@@ -35,10 +35,18 @@ class VariableIndex(NamedTuple):
 
 
 class MonomialIndex(tuple[VariableIndex]):
-    """
+    r"""
     Index for a monomial, i.e. a product of `VariableIndex`.
 
-    `MonomialIndex` has a partial order with respect to the degree.
+    `MonomialIndex` has a total order, they can be sorted with respect to the
+    degree. If they have the same degree then they are sorted with respect to
+    the index of the first variable.
+
+    For example :math:`x^2 z \succeq xy` because the former has degree 3 while
+    the latter is quadratic. Then :math:`yz \succeq xy` (assuming :math:`x` has
+    index 0, :math:`y` has index 1 and :math:`z` 2) because :math:`y \succeq x`.
+
+    Constant monomial indices are considered equal.
     """
 
     @property
@@ -50,7 +58,15 @@ class MonomialIndex(tuple[VariableIndex]):
         return sum(v.power for v in self)
 
     def __lt__(self, other):
-        """ Monomial indices can be sorted with respect to the exponent value. """
+        # Constants are equal
+        if MonomialIndex.is_constant(self):
+            return False
+
+        # they have the same degree the index counts
+        if self.degree == other.degree:
+            # Assumes that monomialindex is sorted!
+            return self[0] < other[0]
+
         return self.degree < other.degree
 
     @staticmethod