Orthogonal decompositions of MV-spaces

Abstract

A maximal disjoint subset $S$ of an $MV$-algebra $A$ is a basis iff $\{x \in A : x \leq a \}$ is a linearly ordered subset of $A$ for all $a \in S$. Let $\Spec A$ be the set of the prime ideals of $A$ with the usual spectral topology. A decomposition $\Spec A = \cup_{i \in I} T_{i} \cup X$ is said to be orthogonal iff each $T_{i}$ is compact open and $S = \{a_{i}\}_{i\in I}$ is a maximal disjoint subset. We prove that this decomposition is unrefinable (i.e. no $T_{i} = \Theta \cap Y$ with $\Theta$ open, $\Theta \cap Y = \emptyset$, int $Y = \emptyset$) iff $S$ is a basis. Many results are established for semisimple $MV$-algebras, which are the algebraic counterpart of Bold fuzzy set theory.