The Stable Topology for MV-algebras

Abstract

Let A be an MV-algebra and I be an ideal of A. Let Spec A be the space of the prime ideals of A with the usual hull-kernel topology. Let U(I) = {P ∈ Spec A/I ⊈ P} be open in Spec A. U(I) is stable open iff Q ∈ Spec A, P ⊆ Q, P ∈ U(I) imply Q ∈ U(I). I is said pure iff U(I) is stable open. Like happens in the commutative rings with unit and in the bounded distributive lattices, the pure ideals of A are characterized via the closed subsets of the hull-kernel topology of Max A, the space of the maximal ideals of A. Opens and stable opens of Max A coincide. Some classes of MV-algebras are also described in terms of their pure ideals.