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Ideals, \(\ell \)-rings and \(\operatorname {MV}^\star \)-algebras. (English) Zbl 1003.06004
For an MV-algebra \(A\) let Rad \(A\) be the intersection of all maximal ideals of \(A\). Put \((\operatorname {Rad} A)^*=\{ x^*: x\in \operatorname {Rad} A\}\). The MV-algebra \(A\) is called perfect if \(A=\operatorname {Rad} A\cup (\operatorname {Rad} A)^*\). The first author and A. Lettieri [Studia Logica 53, 417-432 (1994; Zbl 0812.06010)] defined a functor \(D\) establishing a categorical equivalence between perfect MV-algebras and abelian \(\ell \)-groups. An MV\(^*\)-algebra (denoted also as \(\star \)-algebra) is defined as a perfect algebra \(A\) with a binary operation \(\star \) on the set \(\operatorname {Rad} A\) satisfying certain conditions. If \((A,\star)\) is a \(\star \)-algebra, then by using \(\star \) we can define a multiplication on \(D(A)\) such that \((D(A),\cdot)\) turns out to be an \(\ell \)-ring; this construction gives a categorical equivalence between \(\star \)-algebras and \(\ell \)-rings.
The authors of the present paper prove a series of results on ideals in \(\star \)-algebras and on their connection to the \(\ell \)-ideals in the associated \(\ell \)-rings. From the authors’ introduction: “Section 2 contains some basic notions and results on \(\star \)-ideals in a \(\star \)-algebra. In Section 3 we define \(f\)-algebras, an important class of \(\star \)-algebras corresponding to \(f\)-rings, and in Section 4 we study the \(\star \)-prime ideals in \(f\)-algebras. Section 5 is devoted to some MV-versions of some results of M. Henriksen and S. Larson, and Section 6 to chain conditions in \(f\)-algebras. The paper ends with the investigation of two kinds of reticulations associated with an \(f\)-algebra”.
06D35 MV-algebras
06F25 Ordered rings, algebras, modules
Full Text: EuDML
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