Abstract
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 ⋂ j=1 m cl( coz( f j ))⊂ X is a P-space for every pairwise disjoint family { f j } j=1 m ⊂ C( X).
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