In this paper it is shown that divisibility of a complete lattice ordered (abelian) group is closely related to the existence of a sufficient number of small elements in the positive cone. We shall denote the set of all real numbers by R which symbol will be reserved for this purpose. All terms used are as denned in Birkhoff (1). For the reader's convenience we now define the two terms most used in the sequel. Definition 1. Let (G, ^) be a lattice ordered group. An element ceG is a weak unit if

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... both c>0 and CA \ x | = 0 implies x = 0. Definition 2. A partially ordered group (G, ^) is called integrally closed ifna f£ b for n = 1, 2, ... implies a ^ 0. It is known (Fuchs (2), p. 90) that every complete lattice ordered group is integrally closed. Lemma 1. Let (G, ^) be a lattice ordered group. Assume that there exists a collection S of weak units which has infimum zero. Then ify>0 there exists z>0 such that 0<2z ^ y. Proof. Let 0<2z ^ y. Since (G, ^) is isolated it follows that z>0 and so 02X $, y. It follows that {y-X)Ak = 0 (1) for all such A; for if 0 with e S then 0 in (1). This gives for all 4> e S, (y-y A 4>) A y A (j> = 0, and since is a weak unit, (y-yA)Ay = 0. The left-hand side is (y-) + ^ y we get (y-) + = 0, so 0<y g 0