On central extensions of SL(2,F) admitting left-orderings

For an arbitrary euclidean field F we introduce a central extension (G(F),Φ) of SL(2,F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order-preserving bijections of the convex hull of Q in F. If F=R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2,R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.