{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:58:29Z","timestamp":1760241509770,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,4,23]],"date-time":"2018-04-23T00:00:00Z","timestamp":1524441600000},"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>Relations between I-quasi-valuation maps and ideals in    B C K \/ B C I    -algebras are investigated. Using the notion of an I-quasi-valuation map of a    B C K \/ B C I    -algebra, the quasi-metric space is induced, and several properties are investigated. Relations between the I-quasi-valuation map and the I-valuation map are considered, and conditions for an I-quasi-valuation map to be an I-valuation map are provided. A congruence relation is introduced by using the I-valuation map, and then the quotient structures are established and related properties are investigated. Isomorphic quotient    B C K \/ B C I    -algebras are discussed.<\/jats:p>","DOI":"10.3390\/axioms7020026","type":"journal-article","created":{"date-parts":[[2018,4,24]],"date-time":"2018-04-24T04:44:48Z","timestamp":1524545088000},"page":"26","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Quotient Structures of BCK\/BCI-Algebras Induced by Quasi-Valuation Maps"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2383-664X","authenticated-orcid":false,"given":"Seok-Zun","family":"Song","sequence":"first","affiliation":[{"name":"Department of Mathematics, Jeju National University, Jeju 63243, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3871-217X","authenticated-orcid":false,"given":"Hashem","family":"Bordbar","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Shahid Beheshti University, Tehran 1983969411, Iran"}]},{"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":[[2018,4,23]]},"reference":[{"key":"ref_1","first-page":"19","article-title":"On axiom systems of propositional calculi, XIV","volume":"42","author":"Imai","year":"1966","journal-title":"Proc. Jpn. Acad."},{"key":"ref_2","first-page":"26","article-title":"An algebra related with a propositional calculus","volume":"42","year":"1966","journal-title":"Proc. Jpn. Acad."},{"key":"ref_3","first-page":"125","article-title":"On BCI-algebras","volume":"8","year":"1980","journal-title":"Math. Semin. Notes"},{"key":"ref_4","first-page":"1","article-title":"An introduction to theory of BCK-algebras","volume":"23","author":"Tanaka","year":"1978","journal-title":"Math. Japonica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"859","DOI":"10.5666\/KMJ.2015.55.4.859","article-title":"Quasi-valuation mapd on BCK\/BCI-algebras","volume":"55","author":"Song","year":"2015","journal-title":"Kyungpook Math. J."},{"key":"ref_6","unstructured":"Huang, Y.S. (2006). BCI-Algebra, Science Press."},{"key":"ref_7","unstructured":"Meng, J., and Jun, Y.B. (1994). BCK-Algebras, Kyungmoon Sa Co."},{"key":"ref_8","first-page":"81","article-title":"On atoms in BCC-Algebras","volume":"15","author":"Dudek","year":"1995","journal-title":"Discuss. Math. Algebra Stoch. Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1767","DOI":"10.3233\/JIFS-17006","article-title":"Fuzzy anti-grouped filters and fuzzy normal filters in pseudo BCI-Algebras","volume":"33","author":"Zhang","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"559","DOI":"10.3233\/JIFS-17777","article-title":"Soft set theoretical approach to pseudo BCI-algebras","volume":"34","author":"Zhang","year":"2018","journal-title":"J. Intell. Fuzzy Syst."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/7\/2\/26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:01:44Z","timestamp":1760194904000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/7\/2\/26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,23]]},"references-count":10,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["axioms7020026"],"URL":"https:\/\/doi.org\/10.3390\/axioms7020026","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2018,4,23]]}}}