{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:38:03Z","timestamp":1760402283861,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,9]],"date-time":"2022-01-09T00:00:00Z","timestamp":1641686400000},"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>First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between crossing cubic ideal and commutative crossing cubic ideal is discussed. An example to show that crossing cubic ideal is not commutative crossing cubic ideal is given, and then the conditions in which crossing cubic ideal can be commutative crossing cubic ideal are explored. Characterizations of commutative crossing cubic ideal are discussed, and the relationship between commutative crossing cubic ideal and crossing cubic level set is considered. An extension property of commutative crossing cubic ideal is established, and the translation of commutative crossing cubic ideal is studied. Conditions for the translation of crossing cubic set structure to be commutative crossing cubic ideal are provided, and its characterization is processed.<\/jats:p>","DOI":"10.3390\/axioms11010025","type":"journal-article","created":{"date-parts":[[2022,1,9]],"date-time":"2022-01-09T20:29:26Z","timestamp":1641760166000},"page":"25","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Semigroup Structures and Commutative Ideals of BCK-Algebras Based on Crossing Cubic Set Structures"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1721-1053","authenticated-orcid":false,"given":"Mehmet Ali","family":"\u00d6zt\u00fcrk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Ad\u0131yaman University, Ad\u0131yaman 02040, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6741-8669","authenticated-orcid":false,"given":"Damla","family":"Y\u0131lmaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Erzurum Technical University, Erzurum 25050, Turkey"}]},{"given":"Young Bae","family":"Jun","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,9]]},"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","first-page":"417","article-title":"N-ideals of BCK\/BCI-algebras","volume":"22","author":"Jun","year":"2009","journal-title":"J. Chungcheong Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0165-0114(94)90148-1","article-title":"Rosenfeld\u2019s fuzzy subgroups with interval-valued membership functions","volume":"63","author":"Biswas","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","first-page":"169","article-title":"Fuzzy QS-algebras with interval-valued membership functions","volume":"29","author":"Saeid","year":"2006","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"45","DOI":"10.3923\/jai.2011.45.54","article-title":"A fuzzy inference system for diagnosis of hypothyroidism","volume":"4","author":"Khanale","year":"2011","journal-title":"J. Artif. Intell."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Liu, Z., Qin, K., and Pei, Z. (2017). A method for fuzzy soft sets in decision-making based on an ideal solution. Symmetry, 9.","DOI":"10.3390\/sym9100246"},{"key":"ref_7","first-page":"1791","article-title":"An application of interval-valued fuzzy matrices in medical diagnosis","volume":"5","author":"Meenakshi","year":"2011","journal-title":"Int. J. Math. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1384","DOI":"10.1016\/j.fss.2006.12.018","article-title":"Subsethood, entropy, and cardinality for interval-valued fuzzy sets\u2013An algebraic derivation","volume":"158","author":"Vlachos","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.camwa.2009.04.019","article-title":"Combination of interval-valued fuzzy set and soft set","volume":"58","author":"Yang","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning-I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.3233\/JIFS-18382","article-title":"Cubic Pythagorean fuzzy sets and their application to multi-attribute decision making with unknown weight information","volume":"37","author":"Abbas","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ayaz, T., Al-Shomrani, M.M., Abdullah, S., and Hussain, A. (2020). Evaluation of enterprise production based on spherical cubic hamacher aggregation operators. Mathematics, 8.","DOI":"10.3390\/math8101761"},{"key":"ref_13","unstructured":"Tehreem, Hussain, A., and Khan, M.S.A (2020). Average operators based on spherical cubic fuzzy number and their application in multi-attribute decision making. Ann. Optim. Theory Pract., 3, 83\u2013111."},{"key":"ref_14","first-page":"1","article-title":"Crossing cubic structures as an extension of bipolar fuzzy sets","volume":"22","author":"Jun","year":"2021","journal-title":"Annal. Fuzzy Math. Inform."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"17","DOI":"10.52547\/HATEF.JAHLA.2.1.2","article-title":"Crossing cubic ideals of BCK\/BCI-algebras","volume":"2","author":"Jun","year":"2021","journal-title":"J. Algebr. Hyperstruct. Log. Algebr."},{"key":"ref_16","unstructured":"Meng, J., and Jun, Y.B. (1994). BCK-Algebras, Kyung Moon Sa Co."},{"key":"ref_17","unstructured":"Huang, Y. (2006). BCI-Algebra, Science Press."},{"key":"ref_18","first-page":"49","article-title":"Commutative ideals in BCK-algebras","volume":"9","author":"Meng","year":"1991","journal-title":"Pure Appl. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/1\/25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:27:05Z","timestamp":1760362025000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/1\/25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,9]]},"references-count":18,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["axioms11010025"],"URL":"https:\/\/doi.org\/10.3390\/axioms11010025","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,1,9]]}}}