{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T19:18:07Z","timestamp":1768072687229,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,21]],"date-time":"2019-02-21T00:00:00Z","timestamp":1550707200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009391","name":"University of Tabuk","doi-asserted-by":"publisher","award":["S-0064-1439"],"award-info":[{"award-number":["S-0064-1439"]}],"id":[{"id":"10.13039\/100009391","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A neutrosophic set is initiated by Smarandache, and it is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache. Using this idea, Jun et al. introduced the notion of neutrosophic quadruple     B C K \/ B C I    -numbers, and studied neutrosophic quadruple     B C K \/ B C I    -algebras. As a continuation of Jun et al.\u2019s paper, the notion of implicative neutrosophic quadruple     B C K    -algebras is introduced, and several properties are investigated. Given a set Y, conditions for the neutrosophic quadruple Y-set      N q   ( Y )      to be a neutrosophic quadruple     B C I    -algebra are provided. Conditions for the neutrosophic quadruple Y-set      N q   ( Y )      to be an implicative neutrosophic quadruple     B C K    -algebra are provided. Given subsets I and J of a     B C K    -algebra Y, conditions for the neutrosophic quadruple     ( I , J )    -set      N q   ( I , J )      to be an implicative ideal of the neutrosophic quadruple     B C K    -algebra      N q   ( Y )      are discussed.<\/jats:p>","DOI":"10.3390\/sym11020277","type":"journal-article","created":{"date-parts":[[2019,2,22]],"date-time":"2019-02-22T03:49:44Z","timestamp":1550807384000},"page":"277","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Implicative Neutrosophic Quadruple BCK-Algebras and Ideals"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5596-5841","authenticated-orcid":false,"given":"G.","family":"Muhiuddin","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5559-5951","authenticated-orcid":false,"given":"Ahmad N.","family":"Al-Kenani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P.O. Box 80219, Jeddah 21589, Saudi Arabia"}]},{"given":"Eun Hwan","family":"Roh","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Chinju National University of Education, Jinju 52673, Korea"}]},{"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":[[2019,2,21]]},"reference":[{"key":"ref_1","unstructured":"Smarandache, F. (1998). Neutrosophy, Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning. Available online: http:\/\/fs.gallup.unm.edu\/eBook-neutrosophics6.pdf."},{"key":"ref_2","unstructured":"Smarandache, F. (1999). A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Reserch Press."},{"key":"ref_3","first-page":"287","article-title":"Neutrosophic set-a generalization of the intuitionistic fuzzy set","volume":"24","author":"Smarandache","year":"2005","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2199","DOI":"10.1007\/s10489-017-1070-5","article-title":"Non-linear programming method for multi-criteria decision making problems under interval neutrosophic set environment","volume":"48","author":"Garg","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.cogsys.2018.09.001","article-title":"New logarithmic operational laws and their applications to multiattribute decision making for single-valued neutrosophic numbers","volume":"52","author":"Garg","year":"2018","journal-title":"Cogn. Syst. Res."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Muhiuddin, G., Kim, S.J., and Jun, Y.B. (2019). Implicative N-ideals of BCK-algebras based on neutrosophic N-structures, Discrete Mathematics. Algorithms Appl., in press.","DOI":"10.1142\/S1793830919500113"},{"key":"ref_7","first-page":"36","article-title":"Further results on (\u2208, \u2208)-neutrosophic subalgebras and ideals in BCK\/BCI-algebras","volume":"20","author":"Muhiuddin","year":"2018","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"87","DOI":"10.30948\/afmi.2017.14.1.87","article-title":"Neutrosophic subalgebras of BCK\/BCI-algebras based on neutrosophic points","volume":"14","author":"Saeid","year":"2017","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"75","DOI":"10.30948\/afmi.2017.14.1.75","article-title":"Neutrosophic subalgebras of several types in BCK\/BCI-algebras","volume":"14","author":"Jun","year":"2017","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jun, Y.B., Kim, S.J., and Smarandache, F. (2018). Interval neutrosophic sets with applications in BCK\/BCI-algebra. Axioms, 7.","DOI":"10.3390\/axioms7020023"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Jun, Y.B., Smarandache, F., and Bordbar, H. (2017). Neutrosophic \ud835\udca9-structures applied to BCK\/BCI-algebras. Information, 8.","DOI":"10.3390\/info8040128"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Jun, Y.B., Smarandache, F., Song, S.Z., and Khan, M. (2018). Neutrosophic positive implicative \ud835\udca9-ideals in BCK\/BCI-algebras. Axioms, 7.","DOI":"10.3390\/axioms7010003"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"583","DOI":"10.30948\/afmi.2017.14.6.583","article-title":"Neutrosophic \ud835\udca9-structures and their applications in Semigroups","volume":"14","author":"Khan","year":"2017","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_14","first-page":"1","article-title":"Neutrosophic ideals in BCK\/BCI-algebras based on neutrosophic points","volume":"8","author":"Jun","year":"2018","journal-title":"J. Int. Math. Virtual Inst."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Song, S.Z., Smarandache, F., and Jun, Y.B. (2017). Neutrosophic commutative \ud835\udca9-ideals in BCK-algebras. Information, 8.","DOI":"10.3390\/info8040130"},{"key":"ref_16","first-page":"96","article-title":"Neutrosophic quadruple numbers, refined neutrosophic quadruple numbers, absorbance law, and the multiplication of neutrosophic quadruple numbers","volume":"10","author":"Smarandache","year":"2015","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_17","first-page":"122","article-title":"On neutrosophic quadruple algebraic structures","volume":"12","author":"Akinleye","year":"2016","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jun, Y.B., Song, S.Z., Smarandache, F., and Bordbar, H. (2018). Neutrosophic quadruple BCK\/BCI-algebras. Axioms, 7.","DOI":"10.3390\/axioms7020041"},{"key":"ref_19","unstructured":"Huang, Y. (2006). BCI-Algebra, Science Press."},{"key":"ref_20","unstructured":"Meng, J., and Jun, Y.B. (1994). BCK-Algebras, Kyungmoonsa Co."},{"key":"ref_21","unstructured":"Borzooei, R.A., Takallo, M.M., and Jun, Y.B. (2019). Commutative neutrosophic quadruple ideals of neutrosophic quadruple BCK-algebras. Lobachevskii J. Math., submitted."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/277\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:33:54Z","timestamp":1760186034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/277"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,21]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020277"],"URL":"https:\/\/doi.org\/10.3390\/sym11020277","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,21]]}}}