We consider the lattice-ordered groups Inv(R) and Div(R) of invertible and divisorial fractional ideals of a completely integrally closed Prüfer domain. We prove that Div(R) is the completion of the group Inv(R), and we show there is a faithfully flat extension S of R such that S is a completely integrally closed Bézout domain with Div(R) ∼ = Inv(S). Among the class of completely integrally closed Prüfer domains, we focus on the one-dimensional Prüfer domains. This class includes Dedekind domains, the latter being the one-dimensional Prüfer domains whose maximal ideals are finitely generated. However, numerous interesting examples show that the class of one-dimensional Prüfer domains includes domains that differ quite significantly from Dedekind domains by a number of measures, both group-theoretic (involving Inv(R) and Div(R)) and topological (involving the maximal spectrum of R). We examine these invariants in connection with factorization properties of the ideals of one-dimensional Prüfer domains, putting special emphasis on the class of almost Dedekind domains, those domains for which every localization at a maximal ideal is a rank one discrete valuation domain, as well as the class of SP-domains, those domains for which every proper ideal is a product of radical ideals. For this last class of domains, we show that if in addition the ring has nonzero Jacobson radical, then the lattice-ordered groups Inv(R) and Div(R) are determined entirely by the topology of the maximal spectrum of R, and that the Cantor-Bendixson derivatives of the maximal spectrum reflect the distribution of sharp and dull maximal ideals.

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We discuss a fuzzy result by displaying an example that shows how a classical argument fails to work when one passes from classical logic to fuzzy logic. Precisely, we present an example to show that, in the fuzzy context, the fact that the supremum is naturally used in lieu of the union can alter an argument that may work in the classical context.

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An MV-algebra A is radically principal if every prime ideal P of A is radically principal, i.e., there exists a principal ideal I of A such that Rad ( P ) = Rad ( I ) $ \text{Rad}(P)=\text{Rad}(I) $ . We investigate radically principal MV-algebras and provide some characterizations as well as some classes of examples. We prove a Cohen-like theorem, precisely, an MV-algebra is radically principal if and only if every maximal ideal is radically principal. It is also shown that the radically principal hyperarchemedian MV-algebras are the weakly finite ones and the radically principal Boolean algebras are the finite ones. Radically principal MV-algebras are also studied from the perspective of lattice-ordered groups.