{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:13:00Z","timestamp":1760242380027,"version":"build-2065373602"},"reference-count":59,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2017,6,10]],"date-time":"2017-06-10T00:00:00Z","timestamp":1497052800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The purpose of the paper is to introduce, using the known results concerning the entropy in product MV algebras, the concepts of mutual information and Kullback\u2013Leibler divergence for the case of product MV algebras and examine algebraic properties of the proposed measures. In particular, a convexity of Kullback\u2013Leibler divergence with respect to states in product MV algebras is proved, and chain rules for mutual information and Kullback\u2013Leibler divergence are established. In addition, the data processing inequality for conditionally independent partitions in product MV algebras is proved.<\/jats:p>","DOI":"10.3390\/e19060267","type":"journal-article","created":{"date-parts":[[2017,6,12]],"date-time":"2017-06-12T10:27:59Z","timestamp":1497263279000},"page":"267","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Kullback\u2013Leibler Divergence and Mutual Information of Partitions in Product MV Algebras"],"prefix":"10.3390","volume":"19","author":[{"given":"Dagmar","family":"Markechov\u00e1","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, A. Hlinku 1, SK-949 01 Nitra, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Beloslav","family":"Rie\u010dan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovsk\u00e9ho 40, SK-974 01 Bansk\u00e1 Bystrica, Slovakia"},{"name":"Mathematical Institute, Slovak Academy of Sciences, \u0160tef\u00e1nikova 49, SK-814 73 Bratislava, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,10]]},"reference":[{"key":"ref_1","unstructured":"Gray, R.M. (2009). Entropy and Information Theory, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_3","first-page":"861","article-title":"New Metric Invariant of Transitive Dynamical Systems and Automorphisms of Lebesgue Spaces","volume":"119","author":"Kolmogorov","year":"1958","journal-title":"Dokl. Russ. Acad. Sci."},{"key":"ref_4","unstructured":"Sinai, Y.G. (1990). Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics, Springer."},{"key":"ref_5","first-page":"768","article-title":"On the Notion of Entropy of a Dynamical System","volume":"124","author":"Sinai","year":"1959","journal-title":"Dokl. Russ. Acad. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0165-0114(92)90350-D","article-title":"The entropy of fuzzy dynamical systems and generators","volume":"48","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0165-0114(85)90093-4","article-title":"Probability of fuzzy events defined as denumerable additive measure","volume":"17","author":"Piasecki","year":"1985","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1080\/03081079108945015","article-title":"The Bayes principle and the entropy on fuzzy probability spaces","volume":"20","author":"Mesiar","year":"1991","journal-title":"Int. J. Gen. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0165-0114(97)00024-9","article-title":"Entropy of Fuzzy Partitions\u2014A General Model","volume":"99","author":"Mesiar","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0165-0114(94)00245-3","article-title":"Entropy of a fuzzy dynamical system","volume":"70","author":"Dumitrescu","year":"1995","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.fss.2013.02.006","article-title":"On local entropy of fuzzy partitions","volume":"234","author":"Rahimi","year":"2014","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","first-page":"1","article-title":"A Local Approach to Yager Entropy of Dynamical Systems","volume":"1","author":"Rahimi","year":"2015","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0165-0114(00)00017-8","article-title":"m-Equivalence, entropy and F-dynamical systems","volume":"121","author":"Srivastava","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Markechov\u00e1, D., and Rie\u010dan, B. (2016). Entropy of Fuzzy Partitions and Entropy of Fuzzy Dynamical Systems. Entropy, 18.","DOI":"10.3390\/e18010019"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1007\/s00500-003-0285-z","article-title":"An entropy construction inspired by fuzzy sets","volume":"7","year":"2003","journal-title":"Soft Comput."},{"key":"ref_16","first-page":"135","article-title":"On a type of entropy of dynamical systems","volume":"1","year":"1992","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_17","first-page":"157","article-title":"On some modifications of the entropy of dynamical systems","volume":"42","year":"1994","journal-title":"Atti Semin. Mat. Fis. dell\u2019Univ. Modena"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/0020-0255(85)90027-1","article-title":"A review of fuzzy set aggregation connectives","volume":"36","author":"Dubois","year":"1985","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"339","DOI":"10.14311\/NNW.2013.23.021","article-title":"Entropy and mutual information of experiments in the fuzzy case","volume":"23","year":"2013","journal-title":"Neural Netw. World"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_22","unstructured":"Kullback, S. (1959). Information Theory and Statistics, John Wiley & Sons."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1103\/RevModPhys.48.571","article-title":"Network theory of microscopic and macroscopic behavior of master equation systems","volume":"48","author":"Schnakenberg","year":"1976","journal-title":"Rev. Mod. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Risken, H. (1984). The Fokker-Planck Equation, Methods of Solution and Applications, Springer.","DOI":"10.1007\/978-3-642-96807-5"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Qian, H. (arXiv, 2001). Relative Entropy: Free Energy Associated with Equilibrium Fluctuations and Nonequilibrium Deviations, arXiv.","DOI":"10.1103\/PhysRevE.63.042103"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ellis, R.S. (1985). Entropy, Large Deviations, and Statistical Mechanics, Springer.","DOI":"10.1007\/978-1-4613-8533-2"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Markechov\u00e1, D. (2017). Kullback\u2013Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case. Axioms, 6.","DOI":"10.3390\/axioms6010005"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1090\/S0002-9947-1958-0094302-9","article-title":"Algebraic analysis of many valued logics","volume":"88","author":"Chang","year":"1958","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_29","unstructured":"Pap, E. (2002). Probability on MV-algebras. Handbook of Measure Theory, Elsevier."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Rie\u010dan, B., and Neubrunn, T. (1997). Integral, Measure and Ordering, Springer.","DOI":"10.1007\/978-94-015-8919-2"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Dvure\u010denskij, A., and Pulmannov\u00e1, S. (2000). New Trends in Quantum Structures, Springer.","DOI":"10.1007\/978-94-017-2422-7"},{"key":"ref_32","unstructured":"Mundici, D. (2007, May 26). MV Algebras: A Short Tutorial. Available online: http:\/\/www.matematica.uns.edu.ar\/IXCongresoMonteiro\/Comunicaciones\/Mundici_tutorial.pdf."},{"key":"ref_33","first-page":"889","article-title":"Interpretation of AFC*-algebras in Lukasiewicz sentential calculus","volume":"56","author":"Mundici","year":"1986","journal-title":"J. Funct. Anal."},{"key":"ref_34","first-page":"161","article-title":"Entropy on Effect Algebras with the Riesz Decomposition Property II: MV-Algebras","volume":"41","author":"Manara","year":"2005","journal-title":"Kybernetika"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1007\/s10773-005-7080-9","article-title":"Kolmogorov\u2013Sinaj entropy on MV-algebras","volume":"44","year":"2005","journal-title":"Int. J. Theor. Phys."},{"key":"ref_36","first-page":"21","article-title":"D-posets","volume":"44","author":"Chovanec","year":"1994","journal-title":"Math. Slovaca"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1007\/s10773-007-9457-4","article-title":"Quasiproduct on Boolean D-posets","volume":"47","year":"2008","journal-title":"Int. J. Theor. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"545","DOI":"10.2478\/s12175-014-0224-8","article-title":"On D-posets of fuzzy sets","volume":"64","year":"2014","journal-title":"Math. Slovaca"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1007\/BF02283036","article-title":"Effect algebras and unsharp quantum logics","volume":"24","author":"Foulis","year":"1994","journal-title":"Found. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3092","DOI":"10.1007\/s10773-009-0162-3","article-title":"Probability domains","volume":"49","year":"2010","journal-title":"Int. J. Theor. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"4386","DOI":"10.1007\/s10773-015-2594-2","article-title":"Generalized random events","volume":"54","year":"2015","journal-title":"Int. J. Theor. Phys."},{"key":"ref_42","first-page":"143","article-title":"Entropy on Effect Algebras with the Riesz Decomposition Property I: Basic Properties","volume":"41","author":"Manara","year":"2005","journal-title":"Kybernetika"},{"key":"ref_43","first-page":"20","article-title":"Entropy of Countable Partitions on effect Algebras with the Riesz Decomposition Property and Weak Sequential Effect Algebras","volume":"12","author":"Giski","year":"2015","journal-title":"Cankaya Univ. J. Sci. Eng."},{"key":"ref_44","first-page":"137","article-title":"The Concept of Entropy on D-posets","volume":"10","author":"Ebrahimi","year":"2013","journal-title":"Cankaya Univ. J. Sci. Eng."},{"key":"ref_45","first-page":"143","article-title":"On the product MV-algebras","volume":"16","year":"1999","journal-title":"Tatra Mt. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1023\/A:1008322226835","article-title":"An algebraic approach to propositional fuzzy logic","volume":"9","author":"Montagna","year":"2000","journal-title":"J. Log. Lang. Inf."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1023\/B:CMAJ.0000027234.36585.2c","article-title":"On product MV algebras","volume":"52","year":"2002","journal-title":"Czech. Math J."},{"key":"ref_48","first-page":"193","article-title":"Product MV-algebras","volume":"6","year":"2001","journal-title":"Mult. Valued Log."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s005000050080","article-title":"On the entropy of partitions in product MV-algebras","volume":"4","year":"2000","journal-title":"Soft Comput."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/S0165-0114(00)00024-5","article-title":"On the entropy of dynamical systems in product MV-algebras","volume":"121","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Atanassov, K. (1999). Intuitionistic Fuzzy Sets: Theory and Applications, Physica Verlag.","DOI":"10.1007\/978-3-7908-1870-3"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0165-0114(89)90215-7","article-title":"More on intuitionistic fuzzy sets","volume":"33","author":"Atanassov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_54","first-page":"145","article-title":"On two operations over intuitionistic fuzzy sets","volume":"2","author":"Atanassov","year":"2006","journal-title":"J. Appl. Math. Stat. Inform."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Rie\u010dan, B. (2007). Probability theory on IF events. Algebraic and Proof-Theoretic Aspects of Non-Classical Logics, Springer. Papers in Honor of Daniele Mundici on the Occasion of his 60th Birthday; Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-540-75939-3_17"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Farnoosh, R., Rahimi, M., and Kumar, P. (2016, January 24\u201329). Removing noise in a digital image using a new entropy method based on intuitionistic fuzzy sets. Proceedings of the International Conference on Fuzzy Systems, Vancouver, BC, Canada.","DOI":"10.1109\/FUZZ-IEEE.2016.7737843"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0165-0114(96)84611-2","article-title":"Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets","volume":"78","author":"Burillo","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_58","first-page":"30","article-title":"Entropy on IF-events","volume":"13","year":"2007","journal-title":"Notes Intuit. Fuzzy Sets"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/S0165-0114(98)00402-3","article-title":"Entropy for intuitionistic fuzzy sets","volume":"118","author":"Szmidt","year":"2001","journal-title":"Fuzzy Sets Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/6\/267\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:38:40Z","timestamp":1760207920000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/6\/267"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,10]]},"references-count":59,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2017,6]]}},"alternative-id":["e19060267"],"URL":"https:\/\/doi.org\/10.3390\/e19060267","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,6,10]]}}}