Monotone σ-complete groups with unbounded refinement.

Abstract. The real line R may be characterized as the unique non atomic directed partially ordered abelian group which is monotone σ-complete (countable increasing bounded sequences have suprema), satisfies the countable refinement property (countable sums Σmam=Σnbn of positive elements have common refinements) and which is linearly ordered. We prove here that the latter condition is not redundant, thus solving an old problem by A. Tarski, by proving that there are many spaces (in particular, of arbitrarily large cardinality) satisfying all above listed axioms except linear ordering. §0. Introduction. The real line R may be characterized up to isomorphism as the unique partially ordered abelian group G satisfying the following properties: G is non atomic (i.e., there are no minimal elements of G+ \{0}), directed (i.e., every element is the difference of two positive elements), monotone σ-complete (i.e., every bounded increasing sequence of elements ha