On the spectrum of countable MV-algebras

Abstract. In this paper we consider MV-algebras and their prime spectrum. We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element, that is, the MV-algebra F ree1 of McNaughton functions from [0, 1] to [0, 1], the continuous, piecewise linear functions with integer coefficients. The construction is heavily based on Mundici equivalence between MV-algebras and lattice ordered abelian groups with the strong unit. Also, we heavily use the fact that two MV-algebras have the same spectrum if and only if their lattice of principal ideals is isomorphic. As an intermediate step we consider the MV-algebra A1 of continuous, piecewise linear functions with rational coefficients. It is known that A1 contains F ree1, and that A1 and F ree1 are equispectral. However, A1 is in some sense easy to work with than F ree1. Now, A1 is still countable. To build an equispectral uncountable MV-algebra A2, we consider certain “almost rational” functions on [0, 1], which are rational in every initial segment of [0, 1], but which can have an irrational limit in 1. We exploit heavily, via Mundici equivalence, the properties of divisible lattice ordered abelian groups, which have an additional structure of vector spaces over the rational field.