‎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 information 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.