Lattice-Ordered Matrix Rings Over the Integers

Abstract

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.

Key Words:

• Lattice order
• Lattice-ordered ring
• Matrix ring

Mathematics Subject Classification:

• Primary 06F25
• Secondary 16W80, 16S50, 13J25

ACKNOWLEDGMENT

Professor Stuart A. Steinberg initiated the study of the topic in this article. The first author is very grateful to him for valuable conversations involving the topic.

Notes

Communicated by I. Swanson.