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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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