{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T16:41:51Z","timestamp":1778776911697,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,18]],"date-time":"2024-11-18T00:00:00Z","timestamp":1731888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271319"],"award-info":[{"award-number":["12271319"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201373"],"award-info":[{"award-number":["12201373"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023-JC-QN-0046"],"award-info":[{"award-number":["2023-JC-QN-0046"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Basic Research Program of Shaanxi","award":["12271319"],"award-info":[{"award-number":["12271319"]}]},{"name":"Natural Science Basic Research Program of Shaanxi","award":["12201373"],"award-info":[{"award-number":["12201373"]}]},{"name":"Natural Science Basic Research Program of Shaanxi","award":["2023-JC-QN-0046"],"award-info":[{"award-number":["2023-JC-QN-0046"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>As new kinds of aggregation functions, overlap functions and semi overlap functions are widely applied to information fusion, approximation reasoning, data classification, decision science, etc. This paper extends the semi-overlap function on [0, 1] to the function on complete lattices and investigates the residual implication derived from it; then it explores the construction of a semi-overlap function on complete lattices and some fundamental properties. Especially, this paper introduces a more generalized concept of the \u2018semi-\u0398-\u039e function\u2019, which innovatively unifies the semi-overlap function and semi-grouping function. Additionally, it provides methods for constructing and characterizing the semi-\u0398-\u039e function. Furthermore, this paper characterizes the semi-overlap function on complete lattices and the semi-\u0398-\u039e function on [0, 1] from an algebraic point of view and proves that the algebraic structures corresponding to the inflationary semi-overlap function, the inflationary semi-\u0398-\u039e function, and residual implications derived by each of them are inflationary MTL algebras. This paper further discusses the properties of inflationary MTL algebra and its relationship with non-associative MTL algebra, and it explores the connections between some related algebraic structures.<\/jats:p>","DOI":"10.3390\/axioms13110799","type":"journal-article","created":{"date-parts":[[2024,11,19]],"date-time":"2024-11-19T11:58:07Z","timestamp":1732017487000},"page":"799","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Semi-Overlap Functions on Complete Lattices, Semi-\u0398-\u039e Functions, and Inflationary MTL Algebras"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-4041-1596","authenticated-orcid":false,"given":"Xingna","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Data Science, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eunsuk","family":"Yang","sequence":"additional","affiliation":[{"name":"Rm 417, Center for Humanities & Social Sciences, Department of Philosophy & Institute of Critical Thinking and Writing, Jeonbuk National University, Jeonju 54896, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaohong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Data Science, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1109\/TFUZZ.2011.2173581","article-title":"Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons","volume":"20","author":"Bustince","year":"2012","journal-title":"IEEE Trans. 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