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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)
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References:
[1] L. P. Belluce: Semisimple algebras of infinite valued logic and bold fuzzy set theory. Canad. J. Math. 38 (1986), 1356-1379. · Zbl 0625.03009
[2] L. P. Belluce: Semi-simple and complete \(MV\)-algebras. Algebra Univ. 29 (1992), 1-9. · Zbl 0756.06005
[3] G. Birkhoff: Lattice theory. Amer. Math. Soc. Colloquium Publ. Vol. 25, Third Edition, Providence, 1967. · Zbl 0153.02501
[4] C. C. Chang: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc.88 (1958), 467-490. · Zbl 0084.00704
[5] C. C. Chang: A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 89 (1959), 74-80. · Zbl 0093.01104
[6] D. Gluschankof: Cyclic ordered groups and \(MV\)-algebras. Czechoslovak Math. J. 43 (1993), 249-263. · Zbl 0795.06015
[7] C. Goffman: Remarks on lattice ordered groups and vector lattices. I. Carathéodory functions. Trans. Amer. Math. Soc.88 (1958), 107-120. · Zbl 0088.02602
[8] J. Jakubík: Cardinal properties of lattice ordered groups. Fundam. Math.74 (1972), 85-98. · Zbl 0259.06015
[9] A. G. Kurosh: Group Theory, Third Edition. Moskva, 1967. · Zbl 0189.30801
[10] D. Mundici: Interpretation of \(AFC^*\)-algebras in Lukasiewicz sentential calculus. Journ. Functional. Anal.65 (1986), 15-63. · Zbl 0597.46059
[11] F. Šik: Über subdirekte Summen geordneter Gruppen. Czech. Math. J. 10 (85) (1960), 400-424. · Zbl 0102.26501
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