{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:02:47Z","timestamp":1760241767962,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,8]],"date-time":"2018-08-08T00:00:00Z","timestamp":1533686400000},"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 article deals with new concepts in a product MV-algebra, namely, with the concepts of R\u00e9nyi entropy and R\u00e9nyi divergence. We define the R\u00e9nyi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovi\u010dov\u00e1 (Soft Comput. 2000, 4, 41\u201344). Moreover, we introduce and study the notion of R\u00e9nyi divergence in a product MV-algebra. It is proven that the Kullback\u2013Leibler divergence of states on a given product MV-algebra introduced by Markechov\u00e1 and Rie\u010dan in (Entropy 2017, 19, 267) can be obtained as the limit of their R\u00e9nyi divergence. In addition, the relationship between the R\u00e9nyi entropy and the R\u00e9nyi divergence as well as the relationship between the R\u00e9nyi divergence and Kullback\u2013Leibler divergence in a product MV-algebra are examined.<\/jats:p>","DOI":"10.3390\/e20080587","type":"journal-article","created":{"date-parts":[[2018,8,9]],"date-time":"2018-08-09T03:33:48Z","timestamp":1533785628000},"page":"587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["R\u00e9nyi Entropy and R\u00e9nyi Divergence in Product MV-Algebras"],"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. 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":[[2018,8,8]]},"reference":[{"key":"ref_1","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. 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