m-Quasinormal f-rings

Given a completely regular topological space X, we wish to determine the poset structure of the root system of prime ideals of the ring C(X) of real-valued continuous functions on X; and vice versa. Here, we describe three measures on the poset of prime subgroups of a lattice-ordered group which determine the arithmetic of the group. Then we show that C(X) has the property that the sum of any m minimal prime ideals is a maximal ideal or the entire ring if and only if the subspace intersection operatorj=1mcl(coz(fj))subset ofX is a P-space for every pairwise disjoint family {fj}j=1msubset ofC(X).