On the scarcity of lattice-ordered matrix algebras II

Abstract:

We correct and complete Weinberg’s classification of the lattice-orders of the matrix ring ${\Bbb Q}_2$ and show that this classification holds for the matrix algebra $F_2$ where $F$ is any totally ordered field. In particular, the lattice-order of $F_2$ 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 $F_2$ with $1>0$. It is also shown, assuming a certain maximum condition, that $(F^+)_n$ is essentially the only lattice-order of the algebra $F_n$ in which the identity element is positive.