The Lattice of Filters of a Pseudo BL-Algebra

In this paper we begin the investigation of filters and congruences. We define the filters of a pseudo BL-algebra A and we denote by F(A) ()(AFn) lattice of all filters (normal filters) of A; we put in evidence some results about these lattices. By using the two distance functions we define two binary relations )(FL=and )(FR=, related to a filter F of A; these two relations are equivalence relations, but they are not congruences. They quotient set )(/FLAand )(/FRAare bounded distributive lattices. We give characterizations for the maximal and prime elements on F(A) ()(AFn) and we prove the prime filter theorem. We characterize the pseudo BL-algebras for which the lattice of filters (normal filters) is a Boolean lattice and the archimedean and hyperarchimedean pseudo BL-algebras.