Let M be a right f-module
over the directed po-ring R (i.e., M is a lattice-ordered R-module that is a subdirect
product of a family of totally ordered R-modules), and let g be a nonzero
element of M. There is a natural one-to-one correspondence between the set of
R-values of g in M and the set of Z-values of g in M. This basic fact enables
one to obtain all of the local structure theory for f-modules that Conrad
[Czechoslovak Math. J. 15 (1965)] has obtained for l-groups. There is, in
addition, the interaction between the two structures. For example, a special
element g has the same value in CR(g), the convex l-submodule generated
by g, that it has in CZ(g). Using this structure theory and the fact that a
special element is basic in a Johnson semisimple f-ring, it is shown that a
finitely-valued Johnson semisimple f-ring is a direct sum of unital l-simple
f-rings.
