Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra

Abstract

The purpose of this study is twofold. For the first part, the entropy of countable partitions on an effect algebra with the Riesz decomposition property is defined. In addition, the lower and upper entropy and the conditional entropy considering a suitable state and transformation functions are introduced. Then, some basic properties of these notions are investigated. In the second part, weak sequential effect algebra is introduced followed by a definition for the entropy of countable partitions on this structure. Furthermore, with the help of appropriate state and transformation functions, the notion of entropy, conditional entropy and relative entropy are introduced. In the final step, some properties of these concepts are studied.

Keywords

• Entropy,Effect algebra,Dynamical system,Sequential effect algebra

References

1. [1] M.K. Bennett and D.J. Foulis, Effect algebras and unsharp quantum logics, Foundation of Physics 24, (1994), 1331- 1352.
2. [2] D. Butnariu and P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer Academic Publisher, (1993).
3. [3] A. Dinola, A. Dvurensku, M. Hycko and C. Manara, Entropy on effect algebras with the Riesz decomposition property I: Basic properties, Kybernetika, 2, (2005), 143-160.
4. [4] D. Dumitrescu, Measure-preserving transformation and the entropy of a fuzzy partition, 13th Linz Seminar on Fuzzy Set Theory, (1991), 25-27.
5. [5] D. Dumitrescu, Hierarchical pattern classification, Fuzzy Sets and Systems, 28, (1988), 145-162.
6. [6] D. Dumitrescu, A note on fuzzy information theory, Stud. Univ. Babes - Bolyai Math, 33, (1988), 65-69.
7. [7] D. Dumitrescu, Fuzzy partitions with the connectives T infinity, S infinity, Fuzzy Sets and Systems, 47, (1992), 193-195.
8. [8] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl, 176, (1993), 359-373.

9. [9] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems, 55, (1993), 169-177.
10. [10] D. Dumitrescu, Fuzzy conditional logic, Fuzzy Sets and Systems, 68, (1994), 171-179.
11. [11] D. Dumitrescu, Entropy of fuzzy dynamical systems, Fuzzy Sets and Systems, 70, (1995), 45-57.
12. [12] A. Dvurecenskij and S. Pulmannova, New trends in quantum structures, Kluwer Acad. Publ.,Dordrecht/Boston/London and Ister Science, Bratislava, 2000.
13. [13] M.Ebrahimi, Generators of probability dynamical systems, Differential Geometry-Dynamical Systems, 8, (2006), 90-97.
14. [14] M.Ebrahimi and N. Mohamadi, The entropy function on an algebraic structure with infinte partition and mpreserving transformation generators, Applied Sciences, 12, (2010), 48-63.
15. [15] M.Ebrahimi and U. Mohamadi, m-Generators of fuzzy Dynamical Systems, Cankaya University journal of Science and Engineering, 9, (2012), 67-182.
16. [16] M.Ebrahimi and B.Mosapour, The concept of entropy on D-posets, cankaya University Journal of Science and Engineering, 10, (2013), 137-151.
17. [17] L. Weihua and W. Junde, A uniqueness problem of the sequence product on operator effect algebra E(H), J. Phys. A: Math. Theor, 42, (2009), 185206-185215 .
18. [18] P. Malicky and B. Riecan, On the entropy of dynamical systems. In: Proc. Conference Ergodic Theory and Related Topics II, Georgenthal 1986, Teubner, Leipzig, (1987), 135-138.
19. [19] E. PAP, Pseudo-additive measures and their applications, In: Handbook of Measure Theory, Vol. I, II, NorthHolland, Amsterdam, (2002), 1403-1468.
20. [20] J. Petroviciova, On the entropy of partitions in product MV algebras, Soft Computing, 4, (2000), 41- 44.
21. [21] J. Petroviciova, On the entropy of dynamical systems in product MV algebras. Fuzzy Sets and Systems, 121, (2001), 347-351.
22. [22] K. Ravindran, On a structure theory of effect algebras, PhD. Thesis, Kansas State University, Manhattan, (1996).
23. [23] Sh. Jun and W. Junde, Not each sequential effect algebra is sharply dominating. Phys. Letter A., 373, (2009), 1708-1712.
24. [24] Sh. Jun and W. Junde, Remarks on the sequential effect algebras,Report. Math. Phys, 63, (2009), 441-446.
25. [25] Sh. Jun and W. Junde, Sequential product on standard effect algebra E(H), J. Phys. A: Math. Theor, 44, (2009).
26. [26] W. Jia-Mei, W. JunDe and Ch. Minhyung, Mutual information and relative entropy of sequential effect algebras, Theor. Phys. (Beijing, China), 54, (2010), 215-218.
27. [27] J. Wang, J. Wu and M. Cho, Entropy of partitions on sequential efect algebras, Communications in Theoretical Physics, 53, (2010), 399-402.
28. [28] Y. Zhao and Z. Ma, Conditional entropy of partitions on quantum logic, Communications in Theoretical Physics, 48, (2007), 11-13.