{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:31Z","timestamp":1760060671608,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T00:00:00Z","timestamp":1757548800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shangluo University Foundation","award":["25SKY002"],"award-info":[{"award-number":["25SKY002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A randomly generated fuzzy matrix refers to a fuzzy matrix in which the values of elements belong to the sample space of a [0,1]-random variable that follows a certain probability distribution. This paper studies the max\u2013min transitive closure of two-type randomly generated fuzzy matrices: Bernoulli and classical probabilistic models. By introducing the concept of superposed fuzzy matrices, we investigate the probability distribution of the transitive closure of randomly generated fuzzy matrices for two probabilistic models. First, we presented the arithmetic operation rules for the superposed fuzzy relations. The expected value of the randomly generated Bernoulli fuzzy matrix transition closure was studied. A direct calculation method for the randomly generated fuzzy matrix transitive closure of the classical probability model was provided. Finally, the errors between the direct calculation method and the traditional transitive closure calculation method were compared.<\/jats:p>","DOI":"10.3390\/axioms14090690","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T10:50:04Z","timestamp":1757587804000},"page":"690","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Max\u2013Min Transitive Closure of Randomly Generated Fuzzy Matrix: Bernoulli and Classical Probabilistic Models"],"prefix":"10.3390","volume":"14","author":[{"given":"Nan","family":"Li","sequence":"first","affiliation":[{"name":"School of Economics and Management, Shangluo University, Shangluo 726000, China"}]},{"given":"Xianfeng","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Application, Shangluo University, Shangluo 726000, China"},{"name":"Engineering Research Center of Qinling Health Welfare Big Data, Universities of Shaanxi Province, Shangluo 726000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0220-9436","authenticated-orcid":false,"given":"Wuniu","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710119, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,11]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1109\/TFUZZ.2002.800654","article-title":"Algorithms for the computation of T-transitive closures","volume":"10","author":"Naessens","year":"2002","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/S0165-0114(00)00062-2","article-title":"An optimal algorithm for computing the max\u2013min transitive closure of a fuzzy similarity matrix","volume":"123","author":"Lee","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","unstructured":"Baier, C., and Katoen, J.P. (2008). Principles of Model Checking, MIT Press."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ma, Z., Li, Z., Li, W., Gao, Y., and Li, X. (2022). Model checking fuzzy computation tree logic based on fuzzy decision processes with cost. Entropy, 24.","DOI":"10.3390\/e24091183"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0004-3702(03)00118-8","article-title":"Fuzzy set and possibility theory-based methods in artificial intelligence","volume":"148","author":"Dubois","year":"2003","journal-title":"Artif. Intell."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1109\/TETCI.2022.3192890","article-title":"Self-learning fuzzy automaton with input and output fuzzy sets for system modelling","volume":"7","author":"Ying","year":"2022","journal-title":"IEEE Trans. Emerg. Top. Comput. Intell."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/TETCI.2023.3300189","article-title":"Self-learning modeling in possibilistic model checking","volume":"8","author":"Liu","year":"2024","journal-title":"IEEE Trans. Emerg. Top. Comput. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.23919\/cje.2021.00.333","article-title":"Model checking computation tree logic over multi-valued decision processes and its reduction techniques","volume":"33","author":"Liu","year":"2024","journal-title":"Chin. J. Electron."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sakurai, J.J., and Napolitano, J. (2020). Modern Quantum Mechanics, Cambridge University Press.","DOI":"10.1017\/9781108587280"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liu, W., Li, Z., and Li, Y. (2024). Quantum Reachability Games. IEEE Trans. Emerg. Top. Comput. Intell., 1\u201315. Available online: https:\/\/ieeexplore.ieee.org\/document\/10584096.","DOI":"10.1109\/TETCI.2024.3419704"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0165-0114(87)90130-8","article-title":"Processor power considerations\u2014An application of fuzzy Markov chains","volume":"21","author":"Kruse","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","first-page":"7","article-title":"Fuzzy probabilities: New approach and applications","volume":"115","author":"Buckley","year":"2012","journal-title":"Physica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2674","DOI":"10.1016\/j.fss.2004.10.023","article-title":"A fuzzy approach to Markov decision processes with uncertain transition probabilities","volume":"157","author":"Kurano","year":"2006","journal-title":"Fuzzy Sets Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0306-4573(84)90067-0","article-title":"Fuzzy probabilities","volume":"20","author":"Zadeh","year":"1984","journal-title":"Inf. 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