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Direct product decomposition of MV-algebras. (English) Zbl 0821.06011
To each MV-algebra \({\mathcal A}= (A; \oplus, *, \neg, 0,1)\) we can assign a lattice \(L({\mathcal A})= (A; \wedge, \vee)\) by defining, for all \(x,y\in A\), \(x\wedge y= \neg (\neg x\vee \neg y)\) and \(x\vee y= (x* \neg y)\oplus y\). If \({\mathcal A}_ 1\) and \({\mathcal A}_ 2\) are MV-algebras such that the lattices \(L({\mathcal A}_ 1)\) and \(L({\mathcal A}_ 2)\) are isomorphic then \({\mathcal A}_ 1\) and \({\mathcal A}_ 2\) need not be isomorphic. In this paper it is shown that there is a bijection between the internal product decompositions of \({\mathcal A}\) and those of the lattice \(L({\mathcal A})\). Complete MV-algebras and polars are also considered.

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