Pure ideals in residuated lattices

Ideals in MV algebras are, by definition, kernels of homomorphism. An ideal is the dual of a filter in some special logical algebras but not in non-regular residuated lattices. Ideals in residuated lattices are defined as natural generalizations of ideals in MV algebras. Spec(L), the spectrum of a residuated lattice L, is the set of all prime ideals of L and it can be endowed with the spectral topology. The main scope of this paper is to characterize Spec(L), called the stable topology. In this paper, we introduce and investigate the notion of pure ideal in residuated lattices and using these ideals we study the related spectral topologies. Also, using the model of MV algebras, for a De Morgan residuated lattice L, we construct the Belluce lattice associated with L. This will provide informations about the pure ideals and the prime ideals space of L. So, in this paper we generalize some results relative to MV algebras to the case of residuated lattices.