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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.

##### MSC:
 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
##### Keywords:
MV-algebra; internal product decompositions; polars
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##### References:
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