Transactions on Fuzzy Sets and Systems (Nov 2023)
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 $Free_1$ 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 $A_1$ of continuous, piecewise linear functions with rational coefficients. It is known that $A_1$ contains $Free_1$, and that $A_1$ and $Free_1$ are equispectral. However, $A_1$ is in some sense easy to work with than $Free_1$. Now, $A_1$ is still countable. To build an equispectral uncountable MV-algebra $A_2$, 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.
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