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Abstract
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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 and 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, equipped with the induced dual
hull-kernel topology, and its pure spectrum are homeomorphic.
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