The pure spectrum of a residuated lattice

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