{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:41:13Z","timestamp":1760197273706,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,9]],"date-time":"2018-08-09T00:00:00Z","timestamp":1533772800000},"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>This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order     \u03b1 ,     where     \u03b1 \u2208 ( 0 , 1 ) \u222a ( 1 , \u221e ) ,     of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order     \u03b1 ,     where     \u03b1 &gt; 1 ,     has the property of sub-additivity. This property allows us to define, for     \u03b1 &gt; 1 ,     the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.<\/jats:p>","DOI":"10.3390\/e20080589","type":"journal-article","created":{"date-parts":[[2018,8,9]],"date-time":"2018-08-09T10:36:31Z","timestamp":1533810991000},"page":"589","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Tsallis Entropy of Product MV-Algebra Dynamical Systems"],"prefix":"10.3390","volume":"20","author":[{"given":"Dagmar","family":"Markechov\u00e1","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, A. 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Entropy and Information Theory, Springer."},{"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","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_5","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1186\/s13662-018-1524-z","article-title":"Logical entropy of dynamical systems","volume":"2018","author":"Ebrahimzadeh","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1093\/jigpal\/jzx022","article-title":"Logical information theory: New foundations for information theory","volume":"25","author":"Ellerman","year":"2017","journal-title":"Log. J. IGPL"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Markechov\u00e1, D., and Rie\u010dan, B. (2016). Logical entropy of fuzzy dynamical systems. Entropy, 18.","DOI":"10.3390\/e18040157"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Markechov\u00e1, D., Mosapour, B., and Ebrahimzadeh, A. (2018). Logical Divergence, Logical Entropy, and Logical Mutual Information in Product MV-Algebras. Entropy, 20.","DOI":"10.3390\/e20020129"},{"key":"ref_9","first-page":"53","article-title":"The concept of logical entropy on D-posets","volume":"1","author":"Mohammadi","year":"2016","journal-title":"J. Algebraic Struct. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/phys-2015-0058","article-title":"Logical entropy of quantum dynamical systems","volume":"14","author":"Ebrahimzadeh","year":"2016","journal-title":"Open Phys."},{"key":"ref_11","first-page":"879","article-title":"Quantum conditional logical entropy of dynamical systems","volume":"36","author":"Ebrahimzadeh","year":"2016","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Markechov\u00e1, D., and Rie\u010dan, B. (2017). Logical entropy and logical mutual information of experiments in the intuitionistic fuzzy case. Entropy, 19.","DOI":"10.3390\/e19080429"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ebrahimzadeh, A., Eslami Giski, Z., and Markechov\u00e1, D. (2017). Logical entropy of dynamical systems\u2014A general model. Mathematics, 5.","DOI":"10.3390\/math5010004"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.3233\/JIFS-162303","article-title":"Conditional logical entropy of fuzzy \u03c3-algebras","volume":"33","author":"Ebrahimzadeh","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1016\/j.indag.2017.06.007","article-title":"An introduction of logical entropy on sequential effect algebra","volume":"28","author":"Ebrahimzadeh","year":"2017","journal-title":"Indag. Math."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/0022-247X(68)90078-4","article-title":"Probability measures of fuzzy events","volume":"23","author":"Zadeh","year":"1968","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","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_19","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_20","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0165-0114(96)00266-7","article-title":"The entropy of fuzzy dynamical systems, general scheme, and generators","volume":"96","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_21","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_22","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_23","first-page":"889","article-title":"Interpretation of AFC*-algebras in \u0141ukasiewicz sentential calculus","volume":"56","author":"Mundici","year":"1986","journal-title":"J. Funct. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mundici, D. (2011). Advanced \u0141ukasiewicz Calculus and MV-Algebras, Springer.","DOI":"10.1007\/978-94-007-0840-2"},{"key":"ref_25","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_26","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_27","unstructured":"Pap, E. (2002). Probability on MV-algebras. Handbook of Measure Theory, Elsevier."},{"key":"ref_28","first-page":"143","article-title":"On the product MV-algebras","volume":"16","year":"1999","journal-title":"Tatra Mt. Math."},{"key":"ref_29","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_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","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_32","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_33","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann\u2013Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_34","first-page":"30","article-title":"Quantification methods of classification processes: Concept of structural alpha-entropy","volume":"3","author":"Havrda","year":"1967","journal-title":"Kybernetika"},{"key":"ref_35","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, Springer."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.physa.2007.02.070","article-title":"Generalized Boltzmann factors and the maximum entropy principle: Entropies for complex systems","volume":"380","author":"Hanel","year":"2007","journal-title":"Phys. A Stat. Mech Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/S0378-4371(01)00353-3","article-title":"Generalized entropies from first principles","volume":"300","author":"Almeida","year":"2001","journal-title":"Phys. A Stat. Mech Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"056125","DOI":"10.1103\/PhysRevE.66.056125","article-title":"Statistical mechanics in the context of special relativity","volume":"66","author":"Kaniadakis","year":"2002","journal-title":"Phys. Rev. E Stat. Nonlin. Soft Matter Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/S0378-4371(02)01018-X","article-title":"Deformed exponentials and logarithms in generalized thermostatistics","volume":"316","author":"Naudts","year":"2002","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1103\/PhysRevE.58.1442","article-title":"Generalized entropy-based criterion for consistent testing","volume":"58","author":"Tsallis","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"R956","DOI":"10.1103\/PhysRevE.49.R956","article-title":"Fractal random walks from a variational formalism for Tsallis entropies","volume":"49","author":"Alemany","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1142\/S0218348X95000473","article-title":"Nonextensive thermostatistics and fractals","volume":"3","author":"Tsallis","year":"1995","journal-title":"Fractals"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0378-4371(03)00042-6","article-title":"Nonextensive Statistical mechanics and economics","volume":"324","author":"Tsallis","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1051\/epn:2005615","article-title":"Long-range memory and nonextensivity in financial markets","volume":"36","author":"Borland","year":"2005","journal-title":"Europhys. News"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.physleta.2006.12.012","article-title":"Model-free stochastic processes studied with q-wavelet-based in formational tools","volume":"364","author":"Zunino","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.physleta.2004.12.090","article-title":"The analysis of VF and VT with wavelet-based Tsallis information measure","volume":"336","author":"Huang","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/S0378-4371(01)00621-5","article-title":"Nonextensive entropy measure of EEG following brain injury from cardiac arrest","volume":"305","author":"Tong","year":"2002","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/S0378-4371(02)01529-7","article-title":"Brain electrical activity analysis using wavelet-based informational tools (II): Tsallis non-extensivity and complexity measures","volume":"320","author":"Rosso","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_49","unstructured":"Kumar, V. (2015). Kapur\u2019s and Tsalli\u2019s Entropies: A Communication System Perspective, LAP LAMBERT Academic Publishing."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"249","DOI":"10.21136\/CMJ.1993.128391","article-title":"Cyclic ordered groups and MV-algebras","volume":"43","author":"Gluschankof","year":"1993","journal-title":"Czechoslovak Math. J."},{"key":"ref_51","first-page":"227","article-title":"Independent axiomatization for MV-algebras","volume":"15","author":"Cattaneo","year":"1998","journal-title":"Tatra Mt. Math."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Koleshko, V.M. (2012). Analysis of Fuzzy Logic Models. Intelligent Systems, InTech.","DOI":"10.5772\/2350"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Anderson, M., and Feil, T. (1988). Lattice Ordered Groups, Kluwer.","DOI":"10.1007\/978-94-009-2871-8"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s005000050082","article-title":"On the probability theory on product MV-algebras","volume":"4","year":"2000","journal-title":"Soft Comput."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Walters, P. (1982). An Introduction to Ergodic Theory, Springer.","DOI":"10.1007\/978-1-4612-5775-2"},{"key":"ref_56","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."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/589\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:17:30Z","timestamp":1760195850000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/589"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,9]]},"references-count":56,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["e20080589"],"URL":"https:\/\/doi.org\/10.3390\/e20080589","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,8,9]]}}}