On the scarcity of lattice-ordered matrix rings

It is well-known that the ring Qn of nxn matrices over a lattice-ordered ring Q may be lattice-ordered by prescribing that a matrix is positive exactly when each of its entries is positive. We conjecture in case Q is the field of rational numbers that this is essentially the only lattice-order of the matrix ring in which the multiplicative identity 1 is positive and settle the conjecture in case n=2. There are however other lattice-orders of Q2 in which 1 is not positive. A complete description of this family is obtained.