Atoms and Dobrakov submeasures in effect algebras. (English) Zbl 1176.03046
Summary: The present paper deals with the study of the notion of Dobrakov submeasure \(m\) defined on an effect algebra \(L\) with values in \([0,\infty)\). We establish the equivalence of the following three properties for a Dobrakov submeasure \(m\) defined on a \(\sigma \)-complete effect algebra \(L\): (i) \(m\) is atomless, (ii) \(m\) has the Saks property, (iii) \(m\) has the Darboux property. We also study the concept of an atom of a function \(\mu\) defined on an effect algebra \(L\) with values in \([0,\infty)\), and finally we prove for a monotone, \(m\)-continuous and semicontinuous function \(\mu\) defined on a \(\sigma\)-complete effect algebra \(L\) that \(\mu\) is non-atomic if and only if \(\mu\) is atomless.
MSC:
| 03G12 | Quantum logic |
| 28C99 | Set functions and measures on spaces with additional structure |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
Keywords:
effect algebra; atomless function; Saks property; Darboux property; atom; non-atomic functionReferences:
| [1] | Avallone, A., Cafiero and Nikodym boundedness theorems in effect algebras, Ital. J. Pure. Appl. Math., 20, 203-214 (2006) · Zbl 1121.28015 |
| [2] | Avallone, A.; Basile, A., On a Marinacci uniqueness theorem for measures, J. Math. Anal. Appl., 286, 2, 378-390 (2003) · Zbl 1052.28011 |
| [3] | Avallone, A.; De Simone, A.; Vitolo, P., Effect algebras and extensions of measures, Bollettino U.M.I., 9-B, 8, 423-444 (2006) · Zbl 1115.28010 |
| [4] | Beltrametti, E. G.; Cassinelli, G., The Logic of Quantum Mechanics (1981), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, MA · Zbl 0491.03023 |
| [5] | Bennet, M. K.; Foulis, D. J., Effect algebras and unsharp quantum logics, Found. Phys., 24, 10, 1331-1352 (1994) · Zbl 1213.06004 |
| [6] | Bennet, M. K.; Foulis, D. J., Phi-symmetric effect algebras, Found. Phys., 25, 1699-1722 (1995) |
| [7] | Bennet, M. K.; Foulis, D. J.; Greechie, R. J., Sums and products of interval algebras, Internat. J. Theoret. Phys., 33, 2114-2136 (1994) · Zbl 0815.06015 |
| [8] | Butnariu, D.; Klement, E. P., Triangular Norm-Based Measures and Games with Fuzzy Coalitions (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0804.90145 |
| [9] | Černek, P., Product of submeasures, Acta Math. Univ. Comenian., 40-41, 301-308 (1982) · Zbl 0552.28004 |
| [10] | Dobrakov, I., On submeasures I, Dissertationes Math., 112, 5-35 (1974) · Zbl 0292.28001 |
| [11] | Dobrakov, I.; Dobrakovova, J., On submeasures II, Math. Slovaca, 30, 65-81 (1980) · Zbl 0428.28001 |
| [12] | Dvurečenskij, A.; Pulmannová, S., Difference posets, effects and quantum measurements, Internat. J. Theoret. Phys., 33, 819-825 (1994) · Zbl 0806.03040 |
| [13] | Dvurečenskij, A.; Pulmannová, S., New Trends in Quantum Structures (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0987.81005 |
| [14] | Epstein, L. G.; Zhang, J., Subjective probabilities on subjectively unambiguous events, Econometrica, 69, 2, 265-306 (2001) · Zbl 1020.91048 |
| [15] | Ghirardato, P.; Marinacci, M., Ambiguity made precise: a comparative foundation, J. Econom. Theory, 102, 251-289 (2002) · Zbl 1019.91015 |
| [16] | Hedlícová, J.; Pulmannová, S., Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenian., 45, 247-279 (1996) · Zbl 0922.06002 |
| [17] | Kalmbach, G., Orthomodular Lattices (1983), Academic Press: Academic Press London · Zbl 0512.06011 |
| [18] | Khare, M., Fuzzy \(\sigma \)-algebras and conditional entropy, Fuzzy Sets and Systems, 102, 287-292 (1999) · Zbl 0935.28011 |
| [19] | Khare, M., The dynamics of \(F\)-quantum spaces, Math. Slovaca, 52, 425-432 (2002) · Zbl 1016.60002 |
| [20] | M. Khare, A.K. Singh, Atoms and a Saks type decomposition in effect algebras, submitted for publication.; M. Khare, A.K. Singh, Atoms and a Saks type decomposition in effect algebras, submitted for publication. · Zbl 1265.81005 |
| [21] | M. Khare, A.K. Singh, Weakly tight functions, their Jordan type decomposition and total variation in effect algebras, submitted for publication.; M. Khare, A.K. Singh, Weakly tight functions, their Jordan type decomposition and total variation in effect algebras, submitted for publication. · Zbl 1143.03033 |
| [22] | Khare, M.; Singh, B., Weakly tight functions and their decomposition, Internat. J. Math. Math. Sci., 18, 2991-2998 (2005) · Zbl 1105.28002 |
| [23] | Kôpka, F.; Chovanec, F., \(D\)-posets of fuzzy sets, Tetra Mount. Math. Publ., 1, 83-87 (1992) · Zbl 0797.04011 |
| [24] | Kôpka, F.; Chovanec, F., \(D\)-posets, Math. Slovaca, 44, 21-34 (1994) · Zbl 0789.03048 |
| [25] | Maharam, D., Finitely additive measures on integers, Sankhya Ser. A, 38, 44-99 (1976) · Zbl 0383.60008 |
| [26] | Martelotti, A., Topological properties of the range of a group-valued finitely additive measure, J. Math. Anal. Appl., 110, 411-424 (1985) · Zbl 0616.28002 |
| [27] | Pap, E., The range of null-additive fuzzy and non-fuzzy measures, Fuzzy Sets and Systems, 65, 105-115 (1994) · Zbl 0844.28013 |
| [28] | Pap, E., Null-additive Set Functions (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrechet · Zbl 0856.28001 |
| [29] | Pulmannová, S., A remark to orthomodular partial algebras, Demonstratio Math., 27, 687-699 (1994) · Zbl 0840.03051 |
| [30] | Riečan, B., On the Dobrakov submeasure on fuzzy sets, Fuzzy Sets and Systems, 151, 635-641 (2005) · Zbl 1061.28013 |
| [31] | Riečanova, Z., Distributive atomic effect algebra, Demonstratio Math., 2, 248-259 (2003) · Zbl 1039.03050 |
| [32] | Riečanova, Z., Modular atomic effect algebra and the extensions of subadditive states, Kybernetika, 40, 4, 459-468 (2004) · Zbl 1249.03120 |
| [33] | Srivastava, P.; Khare, M.; Srivastava, Y. K., \(m\)-equivalance, entropy and \(F\)-dynamical systems, Fuzzy Sets and Systems, 121, 275-283 (2001) · Zbl 0982.37006 |
| [34] | Suzuki, H., Atoms of fuzzy measures and fuzzy integrals, Fuzzy Sets and Systems, 41, 329-342 (1991) · Zbl 0759.28015 |
| [35] | Sykes, S. R., Finite modular effect algebras, Adv. in Appl. Math., 19, 240-250 (1997) · Zbl 0883.03049 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
