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On complete $$MV$$-algebras. (English) Zbl 0841.06010
An element $$x$$ of an $$MV$$-algebra (also called a Wajsberg algebra) is called an $$\alpha$$-atom if the interval $$[0,x ]$$ is a chain of cardinality $$\alpha$$. The algebra $$A$$ is said to be $$\alpha$$-atomic if for every $$y\in A\setminus \{0\}$$ there is an $$\alpha$$-atoms $$x$$ such that $$x\leq y$$. The main results of this paper are the following:
(A) For every $$MV$$-algebra $$A$$ and every cardinal $$\alpha$$: (i) $$A$$ is complete and $$\alpha$$-atomic if and only if it is a direct product of complete $$\alpha$$-atomic linearly ordered $$MV$$-algebras. (ii) If $$\alpha>2$$ then $$A$$ is complete, $$\alpha$$-atomic and linearly ordered if and only if it is of type $$R$$. (iii) If $$A$$ is complete and $$\alpha$$-atomic and $$A\neq \{0\}$$ then either $$\alpha=2$$ or $$\alpha$$ is the cardinality of the continuum.
(B) Let $$A$$ be a complete $$MV$$-algebra. Then $$A$$ is isomorphic to a direct product $$A_1\times A_2\times A_3$$, where $$A_1$$ is atomic, $$A_2$$ is $$c$$-atomic (where $$c$$ is the cardinality of the continuum) and for each $$\alpha$$ there are no $$\alpha$$-atoms in $$A_3$$.

##### MSC:
 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
##### Keywords:
Wajsberg algebra; $$\alpha$$-atom; complete $$MV$$-algebra
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##### References:
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