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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)

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