I. Preliminaries. Throughout this paper po-group will mean partially ordered abelian group. A po-group G is semi-closed if geG and ng >0 for some n>0 implies g >0. G is directed if, whenever gi and g2 are elements of G, there is an element gCG such that g ?gi and g > g2. A subset B of G is lower directed (upper directed) if, whenever a, bCB, there is an element xCB such that x? a and x? b (x>a and x>b). B is a dual ideal of G if bCB and a>b implies aCB. If A is a convex subgroup of G, then a natural order is defined in G/A by setting XCGIA positive if X contains a positive element of G. All quotient structures will be ordered in this manner. For the po-group G, G+= {xCG: x>0}. A po-set S satisfies the Riesz Interpolation Property if, whenever Xl, * ..., xm, yi, * , y, are elements of S and xi?yj for 1_i<m, 1 _jn, then there is an element zGS such that xi<z_yj. Birkhoff [1, p. 328], lists some conditions that are equivalent to the Interpolation Property. The following lemma includes those conditions given by Birkhoff.
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