Lattice-Ordered Matrix Algebras with the Usual Lattice Order

Let R be a unital lattice-ordered algebra over a totally ordered field F and Fn be the n × n(n ≥ 2) matrix algebra over F. It is shown that under certain conditions R contains a lattice-ordered subalgebra which is isomorphic to (Fn, (F + )n). In particular, let (Fn, P) be a lattice-ordered algebra over F with the positive cone P. If a certain element is positive in (Fn, P), then (Fn, P) is isomorphic to (Fn, (F + )n).