{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T10:45:23Z","timestamp":1756809923480,"version":"3.40.5"},"reference-count":20,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"We propose a new class of mathematical structures called(<\/mml:mo>m<\/mml:mi>,<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math>-semirings<\/jats:italic>(which generalize the usual semirings) and describe their basic properties. We define partial ordering and generalize the concepts of congruence, homomorphism, and so forth, for(<\/mml:mo>m<\/mml:mi>,<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math>-semirings. Following earlier work by Rao (2008), we consider systems made up of several components whose failures may cause them to fail and represent the set of such systems algebraically as an(<\/mml:mo>m<\/mml:mi>,<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math>-semiring. Based on the characteristics of these components, we present a formalism to compare the fault-tolerance behavior of two systems using our framework of a partially ordered(<\/mml:mo>m<\/mml:mi>,<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math>-semiring.<\/jats:p>","DOI":"10.1155\/2013\/482391","type":"journal-article","created":{"date-parts":[[2013,3,18]],"date-time":"2013-03-18T17:05:19Z","timestamp":1363626319000},"page":"1-10","source":"Crossref","is-referenced-by-count":5,"title":["(<\/mml:mo>m<\/mml:mi>,<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math>-Semirings and a Generalized Fault-Tolerance Algebra of Systems"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4398-5877","authenticated-orcid":true,"given":"Syed Eqbal","family":"Alam","sequence":"first","affiliation":[{"name":"College of Computers and Information Technology, Taif University, Taif 21974, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shrisha","family":"Rao","sequence":"additional","affiliation":[{"name":"IIIT-Bangalore, Bangalore 560 100, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bijan","family":"Davvaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yazd University, Yazd, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"9","doi-asserted-by":"publisher","DOI":"10.1145\/233557.233559"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1145\/1376804.1376813"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1177\/105971239400200403"},{"year":"1999","key":"21","first-page":"xii+381"},{"issue":"3","key":"30","first-page":"161","volume":"6","year":"2008","journal-title":"Journal of Algebra and Discrete Structures"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02999695"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02999696"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1007\/BF02483081"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1007\/BF02483082"},{"issue":"1","key":"15","first-page":"19","volume":"14","year":"1981","journal-title":"Demonstratio Mathematica"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1080\/00927870802466835"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196709005226"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/s100120170031"},{"key":"20","first-page":"444","volume":"68","year":"1965","journal-title":"Matematicheskii Sbornik"},{"key":"18","first-page":"73","volume":"3","year":"1996","journal-title":"Quasigroups and Related Systems"},{"issue":"1","key":"16","first-page":"132","volume":"2","year":"1995","journal-title":"Quasigroups and Related Systems"},{"key":"24","series-title":"Series in Algebra","first-page":"viii+361","volume":"5","year":"1998"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.42.9.632"},{"key":"7","series-title":"Graduate Texts in Mathematics","first-page":"xvi+276","volume":"78","year":"1981"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1969-0237575-4"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/482391.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/482391.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/482391.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,30]],"date-time":"2016-07-30T09:52:16Z","timestamp":1469872336000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/482391\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":20,"alternative-id":["482391","482391"],"URL":"https:\/\/doi.org\/10.1155\/2013\/482391","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2013]]}}}