Subdirect product decompositions of \(MV\)-algebras. (English) Zbl 0951.06012

As is known, every \(MV\)-algebra \(A\) can be represented by means of an abelian lattice ordered group \(G\) with a strong order unit \(u\). The author shows that the lattices of congruences of \(A\) and \(G\) are isomorphic and uses this fact to describe the connections between subdirect product decompositions of \(A\) and of \(G\). Further, he studies some special types of subdirect product decompositions of \(MV\)-algebras.


06D35 MV-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
Full Text: DOI EuDML


[1] G. Birkhoff: Lattice Theory. Providence, 1967. · Zbl 0153.02501
[2] C. C. Chang: A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 89 (1959), 74-80. · Zbl 0093.01104
[3] R. Cignoli, A. Di Nola, A. Lettieri: Priestley duality and quotient lattices of many-valued algebras. Rendinconti Circ. Matem. Palermo, Serie II 40 (1991), 371-384. · Zbl 0787.06013
[4] D. Gluschankof: Cyclic ordered groups and \(MV\)-algebras. Czechoslovak Math. J. 43 (1993), 249-263. · Zbl 0795.06015
[5] J. Jakubík: Direct product decompositions of \(MV\)-algebras. Czechoslovak Math. J. 44 (1994), 725-739. · Zbl 0821.06011
[6] J. Jakubík: Sequential convergences on \(MV\)-algebras. Czechoslovak Math. J. 45 (1995), 709-726. · Zbl 0845.06009
[7] D. Mundici: Interpretation of \(AFC^*\)-algebras in Łukasiewicz sentential calculus. Journ. Functional. Anal. 65 (1986), 15-63. · Zbl 0597.46059
[8] J Rachůnek: \(DR\ell \)-semigroups and \(MV\)-algebras. Czechoslovak Math. J. 48(123) (1998), 341-372. · Zbl 0952.06014
[9] F. Šik: Über subdirekte Summen geordneter Gruppen. Czechoslovak Math. J. 10(85) (1960), 400-424. · Zbl 0102.26501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.