A Lattice-Ordered Skew Field Is Totally Ordered If Squares Are Positive

Abstract

We show that a lattice-ordered field (not necessarily commutative) is totally ordered if and only if each square is positive, answering a generalized question of Conrad and Dauns [6] in the affirmative. As a consequence, any lattice-ordered skew-field in [5] is totally ordered. Furthermore, we note that each lattice order determined by a pre-positive cone P on a skew-filed F is linearly ordered since F 2 ⊆ P (see [10]).