Direct product factors in GMV-algebras. (English) Zbl 1113.06009

GMV-algebras (also known as pseudo-MV algebras) generalize MV-algebras, the algebras corresponding to Łukasiewicz logic, in that the addition is no longer assumed to be commutative. They are representable by intervals of \(\ell \)-groups.
In this paper, the internal direct product decompositions of a GMV-algebra are described, that is, the direct product decompositions such that each factor is among its ideals. The decompositions of a GMV-algebra are moreover related to those of the representing \(\ell \)-group.
Furthermore, the notion of projectibility is introduced for GMV-algebras in analogy to \(\ell \)-groups, and the polars of projectible GMV-algebras are characterized.


06D35 MV-algebras
06F15 Ordered groups


[1] BIGARD A.-KEIMEL K.-WOLFENSTEIN S.: Groupes et Anneaux Réticulés. Springer Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0384.06022
[2] CHANG C. C.: Algebraic analysis of many valued logic. Trans. Amer. Math. Soc. 88 (1958), 467-490. · Zbl 0084.00704
[3] CIGNOLI R. O. L.-D‘OTTAVIANO I. M. L.-MUNDICI D.: Algebraic Foundation of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht-Boston-London, 2000.
[4] DVUREČENSKIJ A.: Pseudo MV-algebras are intervals in t-groups. J. Aust. Math. Soc. 70 (2002), 427-445. · Zbl 1027.06014
[5] DVUREČENSKIJ A.: States on pseudo MV-algebras. Studia Logica 68 (2001), 301-327. · Zbl 1081.06010
[6] DVUREČENSKIJ A.-PULMANNOVÁ S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht-Boston-London, 2000. · Zbl 0987.81005
[7] GEORGESCU G.-IORGULESCU A.: Pseudo MV-algebras. Mult.-Valued Log. 6 (2001), 95-135. · Zbl 1014.06008
[8] GLASS A. M. W.: Partially Ordered Groups. World Scientific, Singapore-New Jersey-London-Hong Kong, 1999. · Zbl 0933.06010
[9] HÁJEK P.: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. · Zbl 0937.03030
[10] JAKUBÍK J.: Direct product decompositions of pseudo MV-algebras. Arch. Math. (Brno) 37 (2001), 131-142. · Zbl 1070.06003
[11] KOVÁŘ T.: A General Theory of Dually Residuated Lattice Ordered Monoids. Thesis, Palacky Univ., Olomouc, 1996.
[12] KÜHR J.: Ideals of noncommutative DRl-monoids. Czechoslovak Math. J. 55 (2005), 97-111. · Zbl 1081.06017
[13] KÜHR J.: A generalization of GMV-algebras. Mult.-Valued Log.
[14] RACHŮNEK J.: A non-commutative generalization of MV-algebras. Czechoslovak Math. J. 52(127) (2002), 255-273. · Zbl 1012.06012
[15] RACHŮNEK J.: Prime spectra of non-commutative generalizations of MV-algebras. Algebra Universalis 48 (2002), 151-169. · Zbl 1058.06015
[16] RACHŮNEK J.-ŠALOUNOVÁ D.: Direct decompositions of dually residuated lattice ordered monoids. Discuss. Math. Gen. Algebra Appl. 24 (2004), 63-74. · Zbl 1068.06016
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