1 8 M ay 2 00 5 A LATTICE - ORDERED SKEW FIELD IS TOTALLY ORDERED IF SQUARES ARE POSITIVE

A partially ordered group G = (G, +, ≤) is both a (not necessarily Abelian) group (G, +) (with binary operation + and identity element 0, where the inverse of a member a of G is denoted by −a) and a partially ordered set (G, ≤) in which a ≥ b implies that a+ c ≥ b+ c and c+ a ≥ c+ b for a, b, and c in G. An element a of G is positive if a ≥ 0. A partially ordered group (G, +, ≤) is called a lattice-ordered group if the partial order ≥ is a lattice order, that is, if each pair of elements x and y in G have a unique least upper bound x ∨ y and a unique greatest lower bound x∧ y. If (G, +, ≤) is a lattice-ordered group and a belongs to G, then the positive part of a is a = a ∨ 0 (≥ 0), the negative part of a is a− = (−a) ∨ 0 (≥ 0), and the absolute value of a is |a| = a ∨ (−a). It is easily seen that a = a − a− and that |a| = a + a− ≥ 0. A lattice-ordered ring R = (R, +, ·, ≤) is a ring that is partially ordered and has the following properties: (i) (R, + , ≤) is a lattice-ordered Abelian group, and (ii) a ≥ 0 and b ≥ 0 imply that ab ≥ 0. A lattice-ordered field (not necessarily commutative) is a lattice-ordered ring whose underlying ring is a division ring.