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Radical classes of MV-algebras. (English) Zbl 0953.06012
For defining the notion of MV-algebra, several (equivalent) systems of axioms have been used in the literature. In the present paper it is shown that an MV-algebra can be characterized as a bounded distributive lattice with a partial binary operation (partial addition) satisfying certain axioms. In this context, the notion of a substructure of an MV-algebra is defined in a natural way. A nonempty class $$Y$$ of MV-algebras which is closed with respect to isomorphisms will be called a radical class if the following conditions are satisfied: (i) whenever $$\mathcal A_1\in Y$$ and $$\mathcal A_2$$ is a substructure of $$\mathcal A_1$$, then $$\mathcal A_2\in Y$$; (ii) if $$\mathcal B$$ is an MV-algebra and $$\mathcal A_1,\mathcal A_2,\dots , \mathcal A_n$$ are substructures of $$\mathcal B$$ such that $$\mathcal A_i\in Y$$ for $$i=1,2,\dots ,n$$, then $$\mathcal A_1\vee \mathcal A_2\vee \dots \vee \mathcal A_n$$ belongs to $$Y$$. Radical classes of lattice ordered groups have been investigated in several papers. Let $$\mathcal R_a$$ and $$\mathcal R_m$$ be the collection of all radical classes of abelian lattice ordered groups or the collection of all radical classes of MV-algebras, respectively. Both $$\mathcal R_a$$ and $$\mathcal R_m$$ are partially ordered by the class-theoretical inclusion. It is proved that the partially ordered collections $$\mathcal R_a$$ and $$\mathcal R_m$$ are isomorphic.
Reviewer: P.Němec (Praha)

##### MSC:
 06D35 MV-algebras 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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