A Study of Groups in a Partially Ordered Linear Algebra
by
Ralph DeMarr
University of New Mexico, Albuquerque, NM 87131
Coauthors: Donald Beken (University of North Carolina at Pembroke, Pembroke, NC 28372), Taen-Yu Dai (York College of City University of New York, Jamaica, NY 11451)

A partially ordered linear algebra (POLA) is an algebra over the real numbers which is partially ordered subject to certain natural conditions. See the paper of T-Y. Dai, Journal of Math Analysis and Appl, vol. 40 (1972), pp. 649-682. A basic example is the algebra of real matrices of some fixed order n, where the partial order is defined entrywise. We let I denote the multiplicative identity of the POLA. We next let G be a subset of the POLA which is a group under the multiplication defined in the POLA. The identity I of the POLA is the identity of the group G. We may now study G by using both the group properties and the algebraic and order properties of the POLA. The case where the group G is fully (totally) ordered is studied; Dai has obtained results in this case. Beken and DeMarr have obtained results in the case where the group G is nearly fully ordered in the following sense: each X in G is comparable with its inverse. A basic problem is to determine if the group is commutative or nearly commutative in some sense.

Date received: November 9, 1998