Fields of quotients of lattice-ordered domains

Abstract.

It is not known whether the field of fractions of an integral domain with a compatible lattice order has a compatible lattice order that extends the given order on the integral domain. The polynomial ring $\mathbb{R}[x,x^{ - 1} ]$ over the real numbers $\mathbb{R}$ has a natural compatible lattice order, viz, the coordinatewise order ≥. We describe circumstances in which the field of fractions of $(\mathbb{R}[x,x^{ - 1} ], + , \cdot , \geq )$ has no archimedean lattice order that extends ≥.