×

zbMATH — the first resource for mathematics

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”.
MSC:
06D35 MV-algebras
06F25 Ordered rings, algebras, modules
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] ANDERSON F. W.: On f-rings with the ascending chain condition. Proc. Amer. Math. Soc. 13 (1962), 715-721. · Zbl 0111.04303
[2] BELLUCE L. P.: Spectral space and non-commutative rings. Comm. Algebra 19 (1991), 1855-1866. · Zbl 0728.16002
[3] BELLUCE L. P.-DI NOLA A.: Yosida type representation for perfect MV-algebras. Math. Logic Quart. 42 (1996), 551-563. · Zbl 0864.06004
[4] BELLUCE L. P.-DI NOLA A.-GEORGESCU G.: Perfect MV-algebras and l-rings. · Zbl 1031.06009
[5] BIGARD A.-KEIMEL K.-WOLFENSTEIN S.: Groupes et anneaux réticulés. Lecture Notes in Math. 608, Springer-Verlag, New York, 1971.
[6] CIGNOLI R.-D’OTTAVIANO I. M. L.-MUNDICI D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht-Boston-London, 2000. · Zbl 0937.06009
[7] DI NOLA A.-LETTIERI A.: Perfect MV-algebras are equivalent to abelian l-groups. Studia Logica 53 (1994), 417-432. · Zbl 0812.06010
[8] HENRIKSEN M.: Semiprime ideals of f-rings. Sympos. Math. 21 (1977), 401-404. · Zbl 0374.06013
[9] JOHNSTONE P. T.: Stone Spaces. Cambridge Univ. Press, Cambridge, 1982. · Zbl 0499.54001
[10] LARSON S.: Convexity conditions on f-rings. Canad. J. Math. 38 (1986), 48-64. · Zbl 0588.06011
[11] LARSON S.: Pseudoprime l-ideals in a class of f-rings. Proc. Amer. Math. Soc. 104 (1988), 685-692. · Zbl 0691.06010
[12] LARSON S.: Minimal convex extensions and intersections of primary l-ideals in f-rings. J. Algebra 123 (1989), 99-110. · Zbl 0676.06023
[13] LARSON S.: Sums of semiprime, z and l-ideals in a class of f-rings. Proc. Amer. Math. Soc. 109 (1990), 895-901. · Zbl 0702.06012
[14] LARSON S.: Primary l-ideals in a class of f-rings. Comm. Algebra 20 (1992), 2075-2094. · Zbl 0777.06010
[15] LARSON S.: Square dominated l-ideals and l-products and sums of semiprime l-ideals in f-rings. Comm. Algebra 20 (1992), 2095-2112. · Zbl 0777.06011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.