## Good and bad infinitesimals, and states on pseudo MV-algebras.(English)Zbl 1081.06010

Pseudo MV-algebras are a noncommutative generalization of MV-algebras. States on pseudo MV-algebras are analogues of probability measures. In contrast to MV-algebras, there are pseudo MV-algebras having no states. By a crucial result of the second author, pseudo MV-algebras can be represented as intervals of unital $$\ell$$-groups. The authors use this representation to study the problem of existence of states on pseudo MV-algebras by means of the properties of their sets of infinitesimal elements and two kinds of their radicals. Moreover, they give many interesting and important examples of pseudo MV-algebras which describe different classes and varieties of pseudo MV-algebras.

### MSC:

 06D35 MV-algebras 03G12 Quantum logic
Full Text:

### References:

 [1] Baer, R.: Free sums of groups and their generalizations. An analysis of the associative law, Amer. J. Math. 41 (1949), 706–742. · Zbl 0033.34504 [2] Chang, C. C.: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490. · Zbl 0084.00704 [3] Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D.: Algebraic Foundations of Many-Valued Reasoning, Kluwer Acad. Publ., 2000. · Zbl 0937.06009 [4] Darnel, M. R.: Theory of Lattice-Ordered Groups, Marcel Dekker, Inc., New York, 1995. · Zbl 0810.06016 [5] Di Nola, A., Georgescu, G. and Iorgulescu, A., Pseudo-BL-algebras, I, II, Multi Valued Logic 8 (2002), 673–714, 717–750. · Zbl 1028.06007 [6] Dvurečenskij, A.: On partial addition in pseudo MV-algebras, in I. Smeureanu et al. (eds), Proc. Fourth Inter. Symp. on Econ. Inform. (May 6–9, 1999, Bucharest), INFOREC Printing House, Bucharest, 1999, pp. 952–960. · Zbl 0984.06009 [7] Dvurečenskij, A.: States on pseudo MV-algebras, Studia Logica 68 (2001), 301–327. · Zbl 0999.06011 [8] Dvurečenskij, A.: Pseudo MV-algebras are intervals in -groups, J. Austral. Math. Soc. 72 (2002), 427–445. · Zbl 1027.06014 [9] Dvurečenskij, A.: States on unital partially-ordered groups, Kybernetika 38 (2002), 297–318. · Zbl 1265.06052 [10] Dvurečenskij, A. and Pulmannová, S.: New Trends in Quantum Structures, Kluwer Acad. Publ., Dordrecht, Ister Science, Bratislava, 2000. · Zbl 0987.81005 [11] Dvurečenskij, A. and Vetterlein, T.: Pseudoeffect algebras. I. Basic properties, Internat. J. Theor. Phys. 40 (2001), 685–701. · Zbl 0994.81008 [12] Dvurečenskij, A. and Vetterlein, T.: Pseudoeffect algebras. II. Group representations, Internat. J. Theor. Phys. 40 (2001), 703–726. · Zbl 0994.81009 [13] Dvurečenskij, A. and Kalmbach, G.: States on pseudo MV-algebras and the hull-kernel topology, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 131–146. · Zbl 1096.06009 [14] Fuchs, L.: Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. · Zbl 0137.02001 [15] Georgescu, G. and Iorgulescu, A.: Pseudo-MV algebras, Multi Valued Logic 6 (2001), 95–135. · Zbl 1014.06008 [16] Glass, A. M. W.: Partially Ordered Groups, World Scientific, Singapore, 1999. · Zbl 0933.06010 [17] Goodearl, K. R.: Partially Ordered Abelian Groups with Interpolation, Math. Surveys Monographs 20, Amer. Math. Soc., Providence, RI, 1986. · Zbl 0589.06008 [18] Hájek, P.: Observations on noncommutative fuzzy logic, Soft Computing 8 (2003), 38–43. · Zbl 1075.03009 [19] Holland, C.: The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10 (1963), 399–408. · Zbl 0116.02102 [20] Kôpka, F. and Chovanec, F.: D-posets, Math. Slovaca 44 (1994), 21–34. [21] Mundici, D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15–63. · Zbl 0597.46059 [22] Mundici, D.: Averaging the truth-value in Łukasiewicz logic, Studia Logica 55 (1995), 113–127. · Zbl 0836.03016 [23] Rachunek, J.: A noncommutative generalization of MV-algebras, Czechoslovak Math. J. 52 (2002), 255–273. · Zbl 1012.06012
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.