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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”.
MSC:
 06D35 MV-algebras 06F25 Ordered rings, algebras, modules
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References:
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