{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,1]],"date-time":"2025-07-01T06:24:16Z","timestamp":1751351056319},"reference-count":16,"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\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We introduce and study principal mappings between posets which generalize the notion of principal elements in a multiplicative lattice, in particular, the principal ideals of a commutative ring. We also consider some weaker forms of principal mappings such as meet principal, join principal, weak meet principal, and weak join principal mappings which also generalize the corresponding notions on elements in a multiplicative lattice, considered by Dilworth, Anderson and Johnson. The principal mappings between the lattices of powersets and chains are characterized. Finally, for any PID<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>, it is proved that a mapping<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>F<\/mml:mi><mml:mo>:<\/mml:mo><mml:mtext>Idl<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2192<\/mml:mo><mml:mtext>Idl<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is a contractive principal mapping if and only if there is a fixed ideal<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>I<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mtext>Idl<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>such that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>J<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>I<\/mml:mi><mml:mi>J<\/mml:mi><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>J<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mtext>Idl<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. This exploration also leads to some new problems on lattices and commutative rings.<\/jats:p>","DOI":"10.1155\/2014\/754019","type":"journal-article","created":{"date-parts":[[2014,5,4]],"date-time":"2014-05-04T17:08:44Z","timestamp":1399223324000},"page":"1-9","source":"Crossref","is-referenced-by-count":1,"title":["Principal Mappings between Posets"],"prefix":"10.1155","volume":"2014","author":[{"given":"Yuan Ting","family":"Nai","sequence":"first","affiliation":[{"name":"Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616"}]},{"given":"Dongsheng","family":"Zhao","sequence":"additional","affiliation":[{"name":"Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616"}]}],"member":"98","reference":[{"key":"7","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1962.12.481"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01236764"},{"key":"9","year":"1972"},{"key":"14","series-title":"London Mathematical Society Student Texts","volume":"51","year":"2000"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01191494"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02485844"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1029000264"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1976-0404067-2"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1972-0306174-8"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1971-0279080-4"},{"key":"16","series-title":"World Scientific Proceedings Series on Computer Engineering and Information Science","first-page":"573","volume-title":"A generalization of Dilworth's principal elements","volume":"5","year":"2012"},{"key":"6","year":"2002"},{"key":"10","year":"2003"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511542725"},{"issue":"4","key":"15","first-page":"39","volume":"13","year":"2005","journal-title":"General Mathematics"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-97-04042-2"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/754019.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/754019.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/754019.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,29]],"date-time":"2016-07-29T04:12:12Z","timestamp":1469765532000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2014\/754019\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":16,"alternative-id":["754019","754019"],"URL":"https:\/\/doi.org\/10.1155\/2014\/754019","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}