Vol. 19, No. 3, 1966

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On the scarcity of lattice-ordered matrix rings

Elliot Carl Weinberg

Vol. 19 (1966), No. 3, 561–571

It is well-known that the ring Qn of n × n matrices over a lattice-ordered ring Q may be lattice-ordered by prescribing that a matrix is positive exactly when each of its entries is positive. We conjecture in case Q is the field of rational numbers that this is essentially the only lattice-order of the matrix ring in which the multiplicative identity 1 is positive and settle the conjecture in case n = 2. There are however other lattice-orders of Q2 in which 1 is not positive. A complete description of this family is obtained.

Mathematical Subject Classification
Primary: 06.85
Secondary: 16.48
Received: 3 August 1965
Published: 1 December 1966
Elliot Carl Weinberg