The going up and going down theorems in MV-algebras and abelian \(l\)-groups
The well-known going up and going down theorems in commutative ring theory relate the prime ideals of a subring to the prime ideals of the overring. The author shows that the going up theorem holds in all MV-algebras, while this is not true in general for the going down theorem. MV-algebras are the Lindenbaum algebras of infinite-valued Łukasiewicz logic, and are deeply related to AF \(C^*\)-algebras, i.e., limits of finite-dimensional \(C^*\)-algebras. For a comprehensive account see the monograph of \textit{R. Cignoli, I. M. L. D'Ottaviano} and \textit{D. Mundici} [Algebraic foundations of many-valued reasoning (Trends in Logic -- Studia Logica Library 7, Kluwer, Dordrecht) (2000; Zbl 0937.06009)]. The author also shows that the lying over theorem [see \textit{I. Kaplansky}, Commutative rings. 2nd ed. (Univ. Chicago Press, Chicago) (1974; Zbl 0296.13001)] holds in all MV-algebras. Using the present reviewer's categorical equivalence between MV-algebras and abelian lattice-ordered groups with strong units [\textit{D. Mundici}, J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)], the author extends his results to this latter class of structures. His main tool is an appropriate notion of ``localization'' at a prime ideal.
scientific article; zbMATH DE number 1423872