{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T22:22:55Z","timestamp":1780525375373,"version":"3.54.1"},"reference-count":19,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Graduate Higher Achievement Foundation of Northwest University","award":["YC13055"],"award-info":[{"award-number":["YC13055"]}]},{"name":"Graduate Higher Achievement Foundation of Northwest University","award":["YZZ12061"],"award-info":[{"award-number":["YZZ12061"]}]},{"name":"Graduate Independent Innovation Foundation of Northwest University","award":["YC13055"],"award-info":[{"award-number":["YC13055"]}]},{"name":"Graduate Independent Innovation Foundation of Northwest University","award":["YZZ12061"],"award-info":[{"award-number":["YZZ12061"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>H<\/mml:mi><mml:mo>,<\/mml:mo><mml:mo>\u2218<\/mml:mo><mml:mo>,<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mo>\u2218<\/mml:mo><mml:mtext>-<\/mml:mtext><mml:mi>compatibled<\/mml:mi><\/mml:math>regular congruence relation<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>\u03b8<\/mml:mi><\/mml:mrow><\/mml:math>and a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>\u03b8<\/mml:mi><mml:mtext>-<\/mml:mtext><mml:mi>compatibled<\/mml:mi><\/mml:math>inf-Bosbach state<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>H<\/mml:mi><mml:mo>,<\/mml:mo><mml:mo>\u2218<\/mml:mo><mml:mo>,<\/mml:mo><mml:mn>0,<\/mml:mn><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>. By inducing an inf-Bosbach state<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math>on the quotient structure<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math><jats:italic>\/<\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mi>\u03b8<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math>, we show that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math><jats:italic>\/<\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mi>\u03b8<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math>is a bounded commutative BCK-algebra which is categorically equivalent to an MV-algebra. In addition, we introduce the notions of hyper measures (states\/measure morphisms\/state morphisms) on hyper BCK-algebras, and present a relation between hyper state-morphisms and Bosbach states. Then we construct a quotient hyper BCK-algebra<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math><jats:italic>\/<\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mtext>Ker<\/mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>by a reflexive hyper BCK-ideal<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:mtext>Ker<\/mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>. Further, we prove that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math><jats:italic>\/<\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mtext>Ker<\/mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>is a bounded commutative BCK-algebra.<\/jats:p>","DOI":"10.1155\/2014\/397265","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T17:01:43Z","timestamp":1394730103000},"page":"1-7","source":"Crossref","is-referenced-by-count":4,"title":["States and Measures on Hyper BCK-Algebras"],"prefix":"10.1155","volume":"2014","author":[{"given":"Xiao-Long","family":"Xin","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northwest University, Xi'an 710069, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northwest University, Xi'an 710069, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"98","reference":[{"key":"10","year":"1993"},{"issue":"2","key":"9","first-page":"49","volume":"24","year":"1983","journal-title":"Mathematica et Physica"},{"key":"17","first-page":"307","volume":"2","year":"1983","journal-title":"International Journal of Applied Mathematics"},{"key":"20","first-page":"37","volume":"13","year":"1999","journal-title":"Ratio Mathematica"},{"key":"16","first-page":"127","volume":"8","year":"2000","journal-title":"Italian Journal of Pure and Applied Mathematics"},{"issue":"2","key":"3","first-page":"267","volume":"76","year":"2012","journal-title":"International Journal of Pure and Applied Mathematics"},{"key":"25","doi-asserted-by":"crossref","first-page":"5","DOI":"10.7151\/dmgaa.1102","volume":"26","year":"2006","journal-title":"Dscussiones Mathematicae General Algebra and Applications"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171204306186"},{"key":"15","first-page":"97","volume":"55","year":"2002","journal-title":"Scientiae Mathematicae Japonicae"},{"issue":"2","key":"14","first-page":"105","volume":"8","year":"2011","journal-title":"Iranian Journal of Fuzzy Systems"},{"issue":"2","key":"6","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1090\/S0002-9947-1958-0094302-9","volume":"88","year":"1958","journal-title":"Transactions of the American Mathematical Society"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01053035"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-010-0571-5"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2012.07.046"},{"key":"11","first-page":"511","volume":"47","year":"1996","journal-title":"Atti del Seminario Matematico e Fisico Dell'universit\u00e3 di Modena"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2010.03.017"},{"key":"12","year":"2000"},{"issue":"1","key":"5","first-page":"83","volume":"61","year":"2005","journal-title":"Scientiae Mathematicae Japonicae"},{"issue":"6","key":"23","first-page":"889","volume":"31","year":"1986","journal-title":"Mathematica Japonica"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/397265.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/397265.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/397265.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T05:34:12Z","timestamp":1498109652000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/397265\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":19,"alternative-id":["397265","397265"],"URL":"https:\/\/doi.org\/10.1155\/2014\/397265","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}