A note on BL-algebras with internal state

Abstract

    The scope of this paper is to put in evidence some properties of the BL-algebras with internal state.  I introduce the concepts of prime and maximal state-filters, I prove a Prime state-filter theorem(Theorem 4.13) and I characterize the set  Rad_{σ}(A), which represents the intersection of all maximal state-filters of a state BL-algebra (A,σ). Also, I introduce the concepts of simple, semisimple and local state BL-algebras relative to its state-filter set.