This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian \(\ell \)-groups. As a first main result, we show that a topological space X is the prime spectrum of an MV-algebra if and only if X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational polyhedra” (a natural generalization of rational polyhedra) of \([0,1]^Y\) for some set Y. As a second main result we extend our results to Abelian \(\ell \)-groups. That is, a topological space X is the prime spectrum of an Abelian \(\ell \)-group if and only if X is generalized spectral, and the lattice K(X) is a closed epimorphic image of the lattice of “cylinder rational cones” (a generalization of rational cones) in \({{\mathbb {R}}}^Y\) for some set Y. Finally, we axiomatize, in monadic second order logic, the Belluce lattices of free MV-algebras (equivalently, the lattice of cylinder rational polyhedra) of dimension 1, 2 and infinite, and we study the problem of describing Belluce lattices in certain fragments of second order logic.

本文研究了表征那些与MV-代数或Abelian \（\ ell \） -群的素谱同胚的拓扑空间的问题。作为第一个主要结果，我们表明，当且仅当X是谱时，拓扑空间X是MV代数的素谱，并且X的紧凑开放子集的晶格K（X）是X的闭胚形图像。对于某些集合 Y，\（[0,1] ^ Y \）的“圆柱有理多面体”（有理多面体的自然概括）的格。作为第二个主要结果，我们将结果扩展到Abelian \（\ ell \） -groups。也就是说，一个拓扑空间X是阿贝尔的主要光谱\（\ ELL \） -基团，当且仅当X是广义的光谱，晶格ķ（X）是“气缸理性锥体”晶格的闭合epimorphic图像（的一般化\（{{\ mathbb {R}}} ^ Y \）中的某些集合Y的有理锥）。最后，我们用一阶二阶逻辑公理化维数为1、2和无穷大的自由MV-代数的Belluce格（等效地，圆柱有理多面体的格），并且研究了描述贝卢斯格的某些问题。二阶逻辑。