{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:45:59Z","timestamp":1760244359969,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T00:00:00Z","timestamp":1672358400000},"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","20&ZD047"],"award-info":[{"award-number":["12271319","20&ZD047"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Social Science Foundation of China","award":["12271319","20&ZD047"],"award-info":[{"award-number":["12271319","20&ZD047"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>As an extension of interval-valued pseudo t-norms, interval-valued pseudo-overlap functions (IPOFs) play a vital role in solving interval-valued multi-attribute decision making problems. However, their corresponding interval-valued algebraic structure has not been studied yet. On the other hand, with the development of non-commutative (non-associative) fuzzy logic, the study of residuated lattice theory is gradually deepening. Due to the conditions of operators being weakened, the algebraic structures are gradually expanding. Therefore, on the basis of interval-valued residuated lattice theory, we generalize and research the related contents of interval-valued general, residuated, lattice-ordered groupoids. In this paper, the concept of interval-valued, general, residuated, lattice-ordered groupoids is given, and some examples are presented to illustrate the relevance of IPOFs to them. Then, in order to further study them, we propose the notions of expanded, interval-valued, general, residuated lattice-ordered groupoids and expanded triangle algebras, and explain that there is one-to-one correspondence between them through a specific proposition. Some of their properties are also analyzed. Lastly, we show the definitions of the filters on the expanded triangle algebras, and investigate the congruence and quotient structure through them.<\/jats:p>","DOI":"10.3390\/axioms12010042","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T03:08:59Z","timestamp":1672628939000},"page":"42","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Interval-Valued General Residuated Lattice-Ordered Groupoids and Expanded Triangle Algebras"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2320-0884","authenticated-orcid":false,"given":"Xiaohong","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"},{"name":"Shaanxi Joint Laboratory of Artificial Intelligence, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9336-437X","authenticated-orcid":false,"given":"Rong","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,30]]},"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":"135","DOI":"10.1016\/j.ins.2016.04.035","article-title":"Autocratic decision making using group recommendations based on ranking interval type-2 fuzzy sets","volume":"361","author":"Cheng","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4948","DOI":"10.1016\/j.apm.2012.10.006","article-title":"A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets","volume":"37","author":"Zhang","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107249","DOI":"10.1016\/j.asoc.2021.107249","article-title":"A wrapper methodology to learn interval-valued fuzzy rule-based classification systems","volume":"104","author":"Sanz","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1109\/TFUZZ.2014.2336263","article-title":"A compact evolutionary interval-valued fuzzy rule-based classification system for the modeling and prediction of real-world financial applications with imbalanced data","volume":"23","author":"Sanz","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_6","unstructured":"Gasse, B.V., Cornelis, C., Deschrijver, G., and Kerre, E. (2006, January 6). Triangle algebras: Towards an axiomatization of interval-valued residuated lattices. Proceedings of the 5th International Conference on Rough Sets and Current Trends in Computing, Kobe, Japan."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1090\/S0002-9947-1939-1501995-3","article-title":"Residuated lattices","volume":"45","author":"Ward","year":"1939","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_8","unstructured":"H\u00e1jek, P. (2013). Metamathematices of Fuzzy Logic, Springer Science Business Media."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s005000100137","article-title":"Pseudo-t-norms and pseudo-BL algebras","volume":"5","author":"Flondor","year":"2001","journal-title":"Soft Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.fss.2011.02.015","article-title":"A non-associative generalization of H\u00e1jek\u2019s BL-algebras","volume":"178","author":"Botur","year":"2011","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","first-page":"1224","article-title":"The properties of generalized triangle algebras","volume":"3","author":"Kim","year":"2013","journal-title":"J. Math. Comput. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Klement, E.P., Mesiar, R., and Pap, E. (2000). Triangular Norms, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-9540-7"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1016\/j.na.2009.08.033","article-title":"Overlap functions","volume":"72","author":"Bustince","year":"2010","journal-title":"Nonlinear Anal."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhang, X., Liang, R., Bustince, H., Bedregal, B., Fernandez, J., Li, M., and Ou, Q. (2022). Pseudo overlap functions, fuzzy implications and pseudo grouping functions with applications. Axioms, 11.","DOI":"10.3390\/axioms11110593"},{"key":"ref_15","unstructured":"Bedregal, B.R.C., and Takahashi, A. (2005, January 25). Interval t-norms as interval representations of t-norms. Proceedings of the 14th IEEE International Conference on Fuzzy Systems, Reno, NV, USA."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijar.2017.07.001","article-title":"Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions","volume":"90","author":"Bedregal","year":"2017","journal-title":"Int. J. Approx. Reason."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.fss.2017.03.007","article-title":"On interval additive generators of interval overlap functions and interval grouping functions","volume":"323","author":"Qiao","year":"2017","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Liang, R., and Zhang, X. (2022). Interval-valued pseudo overlap functions and application. Axioms, 11.","DOI":"10.3390\/axioms11050216"},{"key":"ref_19","first-page":"603","article-title":"Interval fuzzy connectives and pairs of implications","volume":"4","author":"Kim","year":"2014","journal-title":"J. Math. Comput. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1016\/j.fss.2007.09.003","article-title":"Triangle algebras: A formal logic approach to interval-valued residuated lattices","volume":"159","author":"Cornelis","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.ijar.2008.04.006","article-title":"A characterization of interval-valued residuated lattices","volume":"49","author":"Cornelis","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4141","DOI":"10.1090\/S0002-9947-04-03654-2","article-title":"On the finite embeddability property for residuated ordered groupoids","volume":"357","author":"Blok","year":"2005","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3006","DOI":"10.1016\/j.ins.2010.04.010","article-title":"Filters of residuated lattices and triangle algebras","volume":"180","author":"Deschrijver","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hu, M., and Zhang, X. (2022). On cyclic associative semihypergroups and neutrosophic extended triplet cyclic associative semihypergroups. Mathematics, 10.","DOI":"10.3390\/math10040535"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Zhang, X., and Du, Y. (2022). Left (right) regular and transposition regular semigroups and their structures. Mathematics, 10.","DOI":"10.3390\/math10071021"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Liang, R., and Zhang, X. (2022). Pseudo general overlap functions and weak inflationary pseudo BL-algebras. Mathematics, 10.","DOI":"10.3390\/math10163007"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Zhang, X., Liang, R., and Bedregal, B. (2022). Weak inflationary BL-algebras and filters of inflationary (pseudo) general residuated lattices. Mathematics, 10.","DOI":"10.3390\/math10183394"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wang, J., Zhang, X., and Hu, Q. (2022). Three-way fuzzy sets and their applications (II). Axioms, 11.","DOI":"10.3390\/axioms11100532"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hu, M., Wang, C., Yang, J., Wu, Y., Fan, J., and Jing, B. (2022). Rain rendering and construction of rain vehicle color-24 dataset. Mathematics, 10.","DOI":"10.3390\/math10173210"},{"key":"ref_30","unstructured":"Zhan, J., Wang, J., Ding, W., and Yao, Y. (2022). Three-way behavioral decision making with hesitant fuzzy information systems: Survey and challenges. IEEE\/CAA J. Autom. Sin., 1\u201321."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/42\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:48:39Z","timestamp":1760147319000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/42"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,30]]},"references-count":30,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010042"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010042","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,30]]}}}