eja: add brute-force associativity test.

[sage.d.git] / mjo / eja / eja_algebra.py

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index ad6cde724a0d986e92a3c31a2de8ca0baed9563f..3b64ddd7374d8931f80fdefe4b0fb1734b682995 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -178,7 +178,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
 
         category = MagmaticAlgebras(field).FiniteDimensional()
-        category = category.WithBasis().Unital()
+        category = category.WithBasis().Unital().Commutative()
+
         if associative:
             # Element subalgebras can take advantage of this.
             category = category.Associative()
@@ -422,6 +423,18 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         """
         return "Associative" in self.category().axioms()
 
+    def _is_commutative(self):
+        r"""
+        Whether or not this algebra's multiplication table is commutative.
+
+        This method should of course always return ``True``, unless
+        this algebra was constructed with ``check_axioms=False`` and
+        passed an invalid multiplication table.
+        """
+        return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+                    for i in range(self.dimension())
+                    for j in range(self.dimension()) )
+
     def _is_jordanian(self):
         r"""
         Whether or not this algebra's multiplication table respects the
@@ -429,7 +442,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
         We only check one arrangement of `x` and `y`, so for a
         ``True`` result to be truly true, you should also check
-        :meth:`is_commutative`. This method should of course always
+        :meth:`_is_commutative`. This method should of course always
         return ``True``, unless this algebra was constructed with
         ``check_axioms=False`` and passed an invalid multiplication table.
         """
@@ -439,6 +452,81 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                     for i in range(self.dimension())
                     for j in range(self.dimension()) )
 
+    def _jordan_product_is_associative(self):
+        r"""
+        Return whether or not this algebra's Jordan product is
+        associative; that is, whether or not `x*(y*z) = (x*y)*z`
+        for all `x,y,x`.
+
+        This method should agree with :meth:`is_associative` unless
+        you lied about the value of the ``associative`` parameter
+        when you constructed the algebra.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+            ....:                                  ComplexHermitianEJA,
+            ....:                                  QuaternionHermitianEJA)
+
+        EXAMPLES::
+
+            sage: J = RealSymmetricEJA(4, orthonormalize=False)
+            sage: J._jordan_product_is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by()
+            sage: A._jordan_product_is_associative()
+            True
+
+        ::
+
+            sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+            sage: J._jordan_product_is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by(orthonormalize=False)
+            sage: A._jordan_product_is_associative()
+            True
+
+        ::
+
+            sage: J = QuaternionHermitianEJA(2)
+            sage: J._jordan_product_is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by()
+            sage: A._jordan_product_is_associative()
+            True
+
+        """
+        R = self.base_ring()
+
+        # Used to check whether or not something is zero.
+        epsilon = R.zero()
+        if not R.is_exact():
+            # I don't know of any examples that make this magnitude
+            # necessary because I don't know how to make an
+            # associative algebra when the element subalgebra
+            # construction is unreliable (as it is over RDF; we can't
+            # find the degree of an element because we can't compute
+            # the rank of a matrix). But even multiplication of floats
+            # is non-associative, so *some* epsilon is needed... let's
+            # just take the one from _inner_product_is_associative?
+            epsilon = 1e-15
+
+        for i in range(self.dimension()):
+            for j in range(self.dimension()):
+                for k in range(self.dimension()):
+                    x = self.gens()[i]
+                    y = self.gens()[j]
+                    z = self.gens()[k]
+                    diff = (x*y)*z - x*(y*z)
+
+                    if diff.norm() > epsilon:
+                        return False
+
+        return True
+
     def _inner_product_is_associative(self):
         r"""
         Return whether or not this algebra's inner product `B` is
@@ -455,7 +543,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         if not R.is_exact():
             # This choice is sufficient to allow the construction of
             # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
-            epsilon = 1e-16
+            epsilon = 1e-15
 
         for i in range(self.dimension()):
             for j in range(self.dimension()):
@@ -1775,9 +1863,9 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
 
     In theory, our "field" can be any subfield of the reals::
 
-        sage: RealSymmetricEJA(2, field=RDF)
+        sage: RealSymmetricEJA(2, field=RDF, check_axioms=True)
         Euclidean Jordan algebra of dimension 3 over Real Double Field
-        sage: RealSymmetricEJA(2, field=RR)
+        sage: RealSymmetricEJA(2, field=RR, check_axioms=True)
         Euclidean Jordan algebra of dimension 3 over Real Field with
         53 bits of precision
 
@@ -2043,9 +2131,9 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
 
     In theory, our "field" can be any subfield of the reals::
 
-        sage: ComplexHermitianEJA(2, field=RDF)
+        sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True)
         Euclidean Jordan algebra of dimension 4 over Real Double Field
-        sage: ComplexHermitianEJA(2, field=RR)
+        sage: ComplexHermitianEJA(2, field=RR, check_axioms=True)
         Euclidean Jordan algebra of dimension 4 over Real Field with
         53 bits of precision
 
@@ -2340,9 +2428,9 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
 
     In theory, our "field" can be any subfield of the reals::
 
-        sage: QuaternionHermitianEJA(2, field=RDF)
+        sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True)
         Euclidean Jordan algebra of dimension 6 over Real Double Field
-        sage: QuaternionHermitianEJA(2, field=RR)
+        sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True)
         Euclidean Jordan algebra of dimension 6 over Real Field with
         53 bits of precision