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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 \(A_3\).

06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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