On the structure of Archimedean $f$-rings

It is proved that the Boolean valued representation of a Dedekind complete $f$-ring is either the group of integers with zero multiplication, or the ring of integers, or the additive groups of reals with zero multiplication, or the ring of reals. Correspondingly, the Dedekind completion of an Archimedean $f$-ring admits a decomposition into the direct sum of for polars: singular $\ell$-group and an erased vector lattice, both with zero multiplication, a singular $f$-rings and an erased $f$-algebra. A corollary on a functional representation of universally complete $f$-rings is also given.