Geometric approach to the study of partially- and lattice-ordered rings of matrices
by
Piotr J. Wojciechowski
The University of Texas at El Paso
Coauthors: Jingjing Ma (The University of Texas at El Paso)

Let P be a partial order of the ring Q_n of n ×n matrices. We define a P-invariant cone O to be a positive cone of a partially-ordered Q-vector space Q ×Q × ... ×Q such that for every f from P, f(O) is a subset of O. We prove that every directed partial order P has an n-dimensional P-invariant cone. Of special interest is the case when P is completely determined by some cone O, i.e. when P is equal to the set of all matrices f from Q_n for which f(O) is a subset of O. In this situation we say that P is normal. Every maximal directed partial order P is normal. On the other hand, in Q₂, almost all of the normal partial orders are maximal.

If P is a normal partial order on Q₂ and if its P-invariant cone is a positive cone of a lattice order on the vector space Q×Q, then P is isomorphic to the usual lattice order (Q⁺)₂. Conversely, we can show using these methods that in some situations if P is a lattice order on Q₂ and 1 is positive, then the partial order O is a lattice. These considerations provide an alternative approach to the Weinberg's theorem for 2 ×2 matrices and present a hope in addressing his conjecture in higher dimensions.

Date received: February 10, 2000