‎on the spectrum of countable mv-algebras

In this paper we consider mv-algebras and their prime spectrum. we show that there is an uncountablemv-algebra that has the same spectrum as the free mv-algebra over one element, that is, the mv-algebra free1 ofmcnaughton functions from [0; 1] to [0; 1], the continuous, piecewise linear functions with integer coecients. theconstruction is heavily based on mundici equivalence between mv-algebras and lattice ordered abelian groups withthe strong unit. also, we heavily use the fact that two mv-algebras have the same spectrum if and only if theirlattice of principal ideals is isomorphic. as an intermediate step we consider the mv-algebra a1 of continuous,piecewise linear functions with rational coecients. it is known that a1 contains free1, and that a1 and free1are equispectral. however, a1 is in some sense easy to work with than free1. now, a1 is still countable. to buildan equispectral uncountable mv-algebra a2, we consider certain almost rational functions on [0; 1], which arerational 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, whichhave an additional structure of vector spaces over the rational field.