{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:00:45Z","timestamp":1760241645731,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,6,18]],"date-time":"2018-06-18T00:00:00Z","timestamp":1529280000000},"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>The notion of a neutrosophic quadruple BCK\/BCI-number is considered, and a neutrosophic quadruple BCK\/BCI-algebra, which consists of neutrosophic quadruple BCK\/BCI-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple BCK-algebra and a closed ideal in a neutrosophic quadruple BCI-algebra are studied. Given subsets A and B of a BCK\/BCI-algebra, the set NQ(A,B), which consists of neutrosophic quadruple BCK\/BCI-numbers with a condition, is established. Conditions for the set NQ(A,B) to be a (positive implicative) ideal of a neutrosophic quadruple BCK-algebra are provided, and conditions for the set NQ(A,B) to be a (closed) ideal of a neutrosophic quadruple BCI-algebra are given. An example to show that the set {0\u02dc} is not a positive implicative ideal in a neutrosophic quadruple BCK-algebra is provided, and conditions for the set {0\u02dc} to be a positive implicative ideal in a neutrosophic quadruple BCK-algebra are then discussed.<\/jats:p>","DOI":"10.3390\/axioms7020041","type":"journal-article","created":{"date-parts":[[2018,6,18]],"date-time":"2018-06-18T10:57:11Z","timestamp":1529319431000},"page":"41","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Neutrosophic Quadruple BCK\/BCI-Algebras"],"prefix":"10.3390","volume":"7","author":[{"given":"Young Bae","family":"Jun","sequence":"first","affiliation":[{"name":"Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2383-664X","authenticated-orcid":false,"given":"Seok-Zun","family":"Song","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jeju National University, Jeju 63243, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5560-5926","authenticated-orcid":false,"given":"Florentin","family":"Smarandache","sequence":"additional","affiliation":[{"name":"Mathematics &amp; Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3871-217X","authenticated-orcid":false,"given":"Hashem","family":"Bordbar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,18]]},"reference":[{"key":"ref_1","unstructured":"Smarandache, F. 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BCK-Algebras, Kyungmoonsa Co."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/7\/2\/41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:09:08Z","timestamp":1760195348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/7\/2\/41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,18]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["axioms7020041"],"URL":"https:\/\/doi.org\/10.3390\/axioms7020041","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2018,6,18]]}}}