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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)
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References:
[1] C. C. Chang: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490. · Zbl 0084.00704 · doi:10.2307/1993227
[2] C. C. Chang: A new proof of the completeness of the Łukasiewicz axioms. Trans. Amer. Math. Soc. 93 (1959), 74-80. · Zbl 0093.01104 · doi:10.2307/1993423
[3] R. Cignoli: Complete and atomic algebras of the infinite valued Łukasiewicz logic. Studia Logica 50 (1991), 3-4375-384. · Zbl 0753.03026 · doi:10.1007/BF00370678
[4] L. Fuchs: Partially ordered algebraic systems. Pergamon Press, Oxford, 1963. · Zbl 0137.02001
[5] D. Gluschankof: Cyclic ordered groups and \(MV\)-algebras. Czechoslov. Math. J. 43 (1993), 249-263. · Zbl 0795.06015 · eudml:31338
[6] J. Jakubík: Direct product decompositions of \(MV\)-algebras. Czechoslov. Math. J · Zbl 1070.06003 · emis:journals/AM/01-2/index.htm · eudml:232116
[7] D. Mundici: Interpretation of \(AFC^*\)-algebras in Łukasiewicz sentential calculus. Jour. Functional. Anal. 65 (1986), 15-63. · Zbl 0597.46059 · doi:10.1016/0022-1236(86)90015-7
[8] D. Mundici: \(MV\)-algebras are categorically equivalent to bounded commutative \(BCK\)-algebras. Math. Japonica 31 (1986), 889-894. · Zbl 0633.03066
[9] F. Šik: To the theory of lattice ordered groups. Czechoslov. Math. J. 6 (1956), 1-25.
[10] T. Traczyk: On the variety of bounded commutative \(BCK\)-algebras. Math. Japonica 24 (1979), 238-282. · Zbl 0422.03038
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