Finite-dimensional algebras that do not admit a lattice order
by
Jingjing Ma
University of Houston at Clear Lake

In 1956, G. Birkhoff and R.S. Pierce showed that the field of complex numbers cannot be lattice-ordered as a lattice-ordered algebra over the totally ordered field of real numbers, and the field of rational complex numbers cannot be made into a lattice-ordered ring. Later in 1968, R.A. McHaffey showed that the division algebra of real quaternions cannot be made into a real lattice-ordered algebra. Recently, P. Wojciechowski and the author have obtained some results concerning lattice orders on matrix algebras. Motivated by their work, in this talk we provide some conditions to ensure that a subalgebra of a matrix algebra over a subfield of the totally ordered field of real numbers cannot be lattice-ordered as a lattice-ordered algebra. In particular, we show that the matrix algebra over the field of complex numbers and the matrix algebra over the division algebra of real quaternions cannot be lattice-ordered as a real lattice-ordered algebra. Therefore, a finite-dimensional simple real lattice-ordered algebra is isomorphic to a lattice-ordered matrix algebra over the totally ordered field of real numbers, and hence is completely determined. For instance, if the identity element of a finite-dimentional simple real lattice-ordered algebra is positive, then it is isomorphic to the lattice-ordered matrix algebra over the totally ordered field of real numbers with the positive cone consisting of all matrices in which each entry is nonnegative.

Date received: December 30, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-25.