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Let (G, P) be an /-group and ^(P) be the lattice of convex /-subgroups of (G, P). We say that the /-cone Q is essential over P if '¿(Q) is contained in ^(P). It is shown that for each nonzero x in G and each g-value D of x, there is a P-value C of x containing D and no other ß-value of x. We specialize to those essential extensions for which the above C always depends uniquely on x and D; these are called very essential extensions. We show that if (G, P) is a representable /-group then P is thedoi:10.2307/1995740 fatcat:l357mb2krvcy5l3i62iqlrrugu