Pro-finite MV-spaces (Q1827771)

In his classical 1969 paper on the prime ideal structure of commutative rings, \textit{M. Hochster} [Trans. Am. Math. Soc. 142, 43--60 (1969; Zbl 0184.29401)] showed that if \(X\) is a spectral space then \(X\) is homeomorphic to the space of prime ideals of some commutative ring equipped with the hull-kernel topology. The converse is routine. The hull-kernel topology of prime ideals can be similarly defined in MV-algebras -- but the problem of characterizing the topological spaces that arise as prime ideal spaces of MV-algebras (MV-spaces) is still open. Using the present reviewer's Gamma functor, the problem is the same as for lattice-ordered Abelian groups with strong unit, and is essentially the same for lattice-ordered Abelian groups [see \textit{R. L. O. Cignoli}, \textit{I. M. L. D'Ottaviano} and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning. Kluwer Academic Publishers, Dordrecht (2000; Zbl 0937.06009)]. The authors give a characterization in a very special case: their main result is that an MV-space is profinite iff it is a completely normal dual Heyting space. It is known that MV-spaces need not be profinite.