+ self.rank.set_cache(sum(J.rank() for J in algebras))
+
+ def _monomial_to_generator(self, mon):
+ r"""
+ Convert a monomial index into a generator index.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja()
+
+ TESTS::
+
+ sage: J1 = random_eja(field=QQ, orthonormalize=False)
+ sage: J2 = random_eja(field=QQ, orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: all( J.monomial(m)
+ ....: ==
+ ....: J.gens()[J._monomial_to_generator(m)]
+ ....: for m in J.basis().keys() )
+
+ """
+ # The superclass method indexes into a matrix, so we have to
+ # turn the tuples i and j into integers. This is easy enough
+ # given that the first coordinate of i and j corresponds to
+ # the factor, and the second coordinate corresponds to the
+ # index of the generator within that factor.
+ factor = mon[0]
+ idx_in_factor = mon[1]
+
+ offset = sum( f.dimension()
+ for f in self.cartesian_factors()[:factor] )
+ return offset + idx_in_factor
+
+ def product_on_basis(self, i, j):
+ r"""
+ Return the product of the monomials indexed by ``i`` and ``j``.
+
+ This overrides the superclass method because here, both ``i``
+ and ``j`` will be ordered pairs.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (QuaternionHermitianEJA,
+ ....: RealSymmetricEJA)
+
+ TESTS::
+
+ sage: J1 = RealSymmetricEJA(1,field=QQ)
+ sage: J2 = QuaternionHermitianEJA(1,field=QQ)
+ sage: J = cartesian_product([J1,J2])
+ sage: x = sum(J.gens())
+ sage: x == J.one()
+ True
+ sage: x*x == x
+ True
+
+ """
+ l = self._monomial_to_generator(i)
+ m = self._monomial_to_generator(j)
+ return FiniteDimensionalEJA.product_on_basis(self, l, m)
+
+ def matrix_space(self):
+ r"""
+ Return the space that our matrix basis lives in as a Cartesian
+ product.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(1)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.matrix_space()
+ The Cartesian product of (Full MatrixSpace of 1 by 1 dense
+ matrices over Algebraic Real Field, Full MatrixSpace of 2
+ by 2 dense matrices over Algebraic Real Field)
+
+ """
+ from sage.categories.cartesian_product import cartesian_product
+ return cartesian_product( [J.matrix_space()
+ for J in self.cartesian_factors()] )