  This paper studies a fascinating type of filter in residuated lattices, the
so-called pure filters. A combination of algebraic and topological methods on
the pure filters of a residuated lattice is applied to obtain some new
structural results. The notion of purely-prime filters of a residuated lattice
has been investigated, and a Cohen-type theorem has been obtained. It is shown
that the pure spectrum of a residuated lattice is a compact sober space, and a
Grothendieck-type theorem has been demonstrated. It is proved that the pure
spectrum of a Gelfand residuated lattice is a Hausdorff space, and deduced that
the pure spectrum of a Gelfand residuated lattice is homeomorphic to its usual
maximal spectrum. Finally, the pure spectrum of an mp-residuated lattice is
investigated and verified that a given residuated lattice is mp iff its minimal
prime spectrum is equipped with the induced dual hull-kernel topology, and its
pure spectrum is the same.
