On the scarcity of lattice-ordered matrix algebras II

We correct and complete Weinberg's classification of the lattice-orders of the matrix ring and show that this classification holds for the matrix algebra where is any totally ordered field. In particular, the lattice-order of obtained by stipulating that a matrix is positive precisely when each of its entries is positive is, up to isomorphism, the only lattice-order of with . It is also shown, assuming a certain maximum condition, that is essentially the only lattice-order of the algebra in which the identity element is positive.