{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:19:45Z","timestamp":1772252385045,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T00:00:00Z","timestamp":1516665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms\u2014of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.<\/jats:p>","DOI":"10.3390\/axioms7010008","type":"journal-article","created":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T13:06:51Z","timestamp":1516712811000},"page":"8","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem"],"prefix":"10.3390","volume":"7","author":[{"given":"Christian","family":"Servin","sequence":"first","affiliation":[{"name":"Department of Computer Science and Information Technology, El Paso Community College, El Paso, TX 79915, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerardo","family":"Muela","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1244-1650","authenticated-orcid":false,"given":"Vladik","family":"Kreinovich","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,23]]},"reference":[{"key":"ref_1","unstructured":"Awodey, S. (2010). Category Theory, Oxford University Press."},{"key":"ref_2","first-page":"618","article-title":"Definability in Categories. 2. Semigroups and Ordinals","volume":"1","author":"Kreinovich","year":"1980","journal-title":"Abstr. Papers Present. Am. Math. Soc."},{"key":"ref_3","first-page":"234","article-title":"Definability in Categories. 4. Separable Topological Spaces","volume":"2","author":"Kreinovich","year":"1981","journal-title":"Abstr. Papers Present. Am. Math. Soc."},{"key":"ref_4","unstructured":"Feferman, S., and Sieg, W. (2010). Proofs, Categories and Computations, College Publications."},{"key":"ref_5","first-page":"234","article-title":"Definability in Categories. 5. Categories","volume":"2","author":"Kosheleva","year":"1981","journal-title":"Abstr. Papers Present. Am. Math. Soc."},{"key":"ref_6","first-page":"294","article-title":"Definability in Categories. 7. Ultimateness and Applications","volume":"2","author":"Kosheleva","year":"1981","journal-title":"Abstr. 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Proceedings of the Summaries of the First East European Category Seminar, Predela, Bulgaria."},{"key":"ref_11","first-page":"90","article-title":"In category of sets and relations, it is possible to describe functions in purely category terms","volume":"6","author":"Kreinovich","year":"2015","journal-title":"Eur. Math. J."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Belohlavek, R., Dauben, J.W., and Klir, G.J. (2017). Fuzzy Logic and Mathematics: A Historical Perspective, Oxford University Press.","DOI":"10.1093\/oso\/9780190200015.001.0001"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Mendel, J.M. (2017). Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions, Springer.","DOI":"10.1007\/978-3-319-51370-6"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Nguyen, H.T., and Walker, E.A. (2006). A First Course in Fuzzy Logic, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420057102"},{"key":"ref_15","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_16","doi-asserted-by":"crossref","unstructured":"Klir, G., and Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic, Prentice Hall.","DOI":"10.1109\/45.468220"},{"key":"ref_17","unstructured":"Hazewinkel, M. (2001). Equivalence of categories. Encyclopedia of Mathematics, Springer Science and Business Media."},{"key":"ref_18","unstructured":"Mac Lane, S. (1998). Categories for the Working Mathematician, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kozlowski, J., and Melton, A. (2001). Categories: A free tour. 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