Projective limits of \(MV\)-spaces
An \(MV\)-algebra is an abelian monoid \((B,\oplus,0)\) with an operation \(\neg\) such that \(\neg\neg x=x\), \(x\oplus\neg 0=\neg 0\) and \(\neg(\neg x\oplus y)\oplus y=\neg (\neg y\oplus x) \oplus x\). One usually also defines \(1=\neg 0\). For any abelian lattice-group \(G\) with strong unit \(u\), let \(\Gamma(G,u)= [0,u]=\{x\in G|0\leq x\leq u\}\) equipped with the operations \(\neg x=u-x\), \(x\oplus y=(x+y) \wedge u\). Further, for every morphism \(f:(G,u)\to (G',u')\) let \(\Gamma(f)\) be the restriction of \(f\) to \([0,1]\). Then, as proved by the present reviewer in his paper ``Interpretation of AF \(C^*\)-algebras in Lukasiewicz sentential calculus'' [J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)], \(\Gamma\) is a categorical equivalence between abelian lattice-groups with strong units, and \(MV\) algebras. The primitive spectrum of an \(MV\)-algebra and of its corresponding lattice-group are the same. The topological characterization of the spaces arising as primitive spectra of \(MV\)-algebras (called \(MV\)-spaces by the authors) is an important open problem. The main result of this paper is that any projective limit of \(MV\)-spaces within the category of spectral spaces is still an \(MV\)-space.
scientific article; zbMATH DE number 1044151
scientific article