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.
[1] M.K. Bennett and D.J. Foulis, Effect algebras and unsharp quantum logics, Foundation of Physics 24, (1994),
1331- 1352.
[2] D. Butnariu and P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer Academic
Publisher, (1993).
[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] D. Dumitrescu, Measure-preserving transformation and the entropy of a fuzzy partition, 13th Linz Seminar on
Fuzzy Set Theory, (1991), 25-27.
[5] D. Dumitrescu, Hierarchical pattern classification, Fuzzy Sets and Systems, 28, (1988), 145-162.
[6] D. Dumitrescu, A note on fuzzy information theory, Stud. Univ. Babes - Bolyai Math, 33, (1988), 65-69.
[7] D. Dumitrescu, Fuzzy partitions with the connectives T infinity, S infinity, Fuzzy Sets and Systems, 47, (1992),
193-195.
[8] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl, 176, (1993), 359-373.
[9] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems, 55, (1993), 169-177.
[10] D. Dumitrescu, Fuzzy conditional logic, Fuzzy Sets and Systems, 68, (1994), 171-179.
[11] D. Dumitrescu, Entropy of fuzzy dynamical systems, Fuzzy Sets and Systems, 70, (1995), 45-57.
[12] A. Dvurecenskij and S. Pulmannova, New trends in quantum structures, Kluwer Acad.
Publ.,Dordrecht/Boston/London and Ister Science, Bratislava, 2000.
[13] M.Ebrahimi, Generators of probability dynamical systems, Differential Geometry-Dynamical Systems, 8, (2006),
90-97.
[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] M.Ebrahimi and U. Mohamadi, m-Generators of fuzzy Dynamical Systems, Cankaya University journal of Science
and Engineering, 9, (2012), 67-182.
[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] 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] 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] E. PAP, Pseudo-additive measures and their applications, In: Handbook of Measure Theory, Vol. I, II, NorthHolland,
Amsterdam, (2002), 1403-1468.
[20] J. Petroviciova, On the entropy of partitions in product MV algebras, Soft Computing, 4, (2000), 41- 44.
[21] J. Petroviciova, On the entropy of dynamical systems in product MV algebras. Fuzzy Sets and Systems, 121,
(2001), 347-351.
[22] K. Ravindran, On a structure theory of effect algebras, PhD. Thesis, Kansas State University, Manhattan, (1996).
[23] Sh. Jun and W. Junde, Not each sequential effect algebra is sharply dominating. Phys. Letter A., 373, (2009),
1708-1712.
[24] Sh. Jun and W. Junde, Remarks on the sequential effect algebras,Report. Math. Phys, 63, (2009), 441-446.
[25] Sh. Jun and W. Junde, Sequential product on standard effect algebra E(H), J. Phys. A: Math. Theor, 44, (2009).
[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] J. Wang, J. Wu and M. Cho, Entropy of partitions on sequential efect algebras, Communications in Theoretical
Physics, 53, (2010), 399-402.
[28] Y. Zhao and Z. Ma, Conditional entropy of partitions on quantum logic, Communications in Theoretical Physics,
48, (2007), 11-13.
[1] M.K. Bennett and D.J. Foulis, Effect algebras and unsharp quantum logics, Foundation of Physics 24, (1994),
1331- 1352.
[2] D. Butnariu and P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer Academic
Publisher, (1993).
[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] D. Dumitrescu, Measure-preserving transformation and the entropy of a fuzzy partition, 13th Linz Seminar on
Fuzzy Set Theory, (1991), 25-27.
[5] D. Dumitrescu, Hierarchical pattern classification, Fuzzy Sets and Systems, 28, (1988), 145-162.
[6] D. Dumitrescu, A note on fuzzy information theory, Stud. Univ. Babes - Bolyai Math, 33, (1988), 65-69.
[7] D. Dumitrescu, Fuzzy partitions with the connectives T infinity, S infinity, Fuzzy Sets and Systems, 47, (1992),
193-195.
[8] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl, 176, (1993), 359-373.
[9] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems, 55, (1993), 169-177.
[10] D. Dumitrescu, Fuzzy conditional logic, Fuzzy Sets and Systems, 68, (1994), 171-179.
[11] D. Dumitrescu, Entropy of fuzzy dynamical systems, Fuzzy Sets and Systems, 70, (1995), 45-57.
[12] A. Dvurecenskij and S. Pulmannova, New trends in quantum structures, Kluwer Acad.
Publ.,Dordrecht/Boston/London and Ister Science, Bratislava, 2000.
[13] M.Ebrahimi, Generators of probability dynamical systems, Differential Geometry-Dynamical Systems, 8, (2006),
90-97.
[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] M.Ebrahimi and U. Mohamadi, m-Generators of fuzzy Dynamical Systems, Cankaya University journal of Science
and Engineering, 9, (2012), 67-182.
[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] 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] 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] E. PAP, Pseudo-additive measures and their applications, In: Handbook of Measure Theory, Vol. I, II, NorthHolland,
Amsterdam, (2002), 1403-1468.
[20] J. Petroviciova, On the entropy of partitions in product MV algebras, Soft Computing, 4, (2000), 41- 44.
[21] J. Petroviciova, On the entropy of dynamical systems in product MV algebras. Fuzzy Sets and Systems, 121,
(2001), 347-351.
[22] K. Ravindran, On a structure theory of effect algebras, PhD. Thesis, Kansas State University, Manhattan, (1996).
[23] Sh. Jun and W. Junde, Not each sequential effect algebra is sharply dominating. Phys. Letter A., 373, (2009),
1708-1712.
[24] Sh. Jun and W. Junde, Remarks on the sequential effect algebras,Report. Math. Phys, 63, (2009), 441-446.
[25] Sh. Jun and W. Junde, Sequential product on standard effect algebra E(H), J. Phys. A: Math. Theor, 44, (2009).
[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] J. Wang, J. Wu and M. Cho, Entropy of partitions on sequential efect algebras, Communications in Theoretical
Physics, 53, (2010), 399-402.
[28] Y. Zhao and Z. Ma, Conditional entropy of partitions on quantum logic, Communications in Theoretical Physics,
48, (2007), 11-13.
Giski, Z. E., & Ebrahimi, M. (2015). Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra. Cankaya University Journal of Science and Engineering, 12(1).
AMA
Giski ZE, Ebrahimi M. Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra. CUJSE. May 2015;12(1).
Chicago
Giski, Zahra Eslami, and Mohamad Ebrahimi. “Entropy of Countable Partitions on Effect Algebra With Rieze Decomposition Property and Weak Sequential Effect Algebra”. Cankaya University Journal of Science and Engineering 12, no. 1 (May 2015).
EndNote
Giski ZE, Ebrahimi M (May 1, 2015) Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra. Cankaya University Journal of Science and Engineering 12 1
IEEE
Z. E. Giski and M. Ebrahimi, “Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra”, CUJSE, vol. 12, no. 1, 2015.
ISNAD
Giski, Zahra Eslami - Ebrahimi, Mohamad. “Entropy of Countable Partitions on Effect Algebra With Rieze Decomposition Property and Weak Sequential Effect Algebra”. Cankaya University Journal of Science and Engineering 12/1 (May 2015).
JAMA
Giski ZE, Ebrahimi M. Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra. CUJSE. 2015;12.
MLA
Giski, Zahra Eslami and Mohamad Ebrahimi. “Entropy of Countable Partitions on Effect Algebra With Rieze Decomposition Property and Weak Sequential Effect Algebra”. Cankaya University Journal of Science and Engineering, vol. 12, no. 1, 2015.
Vancouver
Giski ZE, Ebrahimi M. Entropy of Countable Partitions on Effect Algebra with Rieze Decomposition Property and Weak Sequential Effect Algebra. CUJSE. 2015;12(1).