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 (Pacific J. Math. 30 (1969), 385--398) in the affirmative. As a consequence, any lattice-ordered skew field in (Brumfiel, Partially ordered rings and semi-algebraic geometry. Cambridge University Press, 1979) is totally ordered. Furthermore, we note that every lattice order determined by a {\it pre-positive cone} $P$ on a skew-filed $F$ is linearly ordered since $F^2\subseteq P$ (see P restel, Lectures on formally real fields, Lecture Notes in mathematics, 1093, Springer-Verlag, 1984).