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On archimedean \(MV\)-algebras. (English) Zbl 0951.06011
An \(MV\)-algebra \(\mathcal A\) constructed by means of an abelian lattice ordered group \(G\) with a strong unit (\(G\) is uniquely determined by \(\mathcal A\)) is called archimedean (or semisimple) if \(G\) is archimedean. A non-empty subset \(\{a_j\mid j\in J\}\) of \(\mathcal A\) is said to be orthogonal if \(a_i\wedge a_j=0\) for all distinct \(i,j\in J\). We say that \(\mathcal A\) is orthogonally complete if every orthogonal subset of \(\mathcal A\) possesses the supremum in \(\mathcal A\). The author has proven:
An \(MV\)-algebra \(\mathcal A\) is orthogonally complete, archimedean and atomic if and only if \(\mathcal A\) is complete and atomic.
An \(MV\)-algebra \(\mathcal A\) is orthogonally complete, archimedean and \(\alpha \)-atomic if and only if \(\mathcal A\) is a product of linearly ordered \(\alpha \)-atomic \(MV\)-algebras for some cardinal \(\alpha > 1\).
If an \(MV\)-algebra is archimedean and \(\alpha \)-atomic for a cardinal \(\alpha >1\) then \(\alpha \in \{2,\aleph _0,c\}\) where \(c\) is the cardinality of the continuum.
Any archimedean orthogonally complete \(MV\)-algebra \(\mathcal A\) is isomorphic to a product \(\mathcal A_1\times \mathcal A_2\times \mathcal A_3\times \mathcal A_4\) of \(MV\)-algebras such that \(\mathcal A_1\) is atomic, \(\mathcal A_2\) is \(\aleph _0\)-atomic, \( \mathcal A_3\) is \(c\)-atomic, and \(\mathcal A_4\) has no \(\alpha \)-atom for any cardinal \(\alpha >1\).
Several examples are presented.

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