{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,15]],"date-time":"2024-09-15T22:53:33Z","timestamp":1726440813327},"reference-count":0,"publisher":"Wiley","issue":"53","license":[{"start":{"date-parts":[[2000,1,1]],"date-time":"2000-01-01T00:00:00Z","timestamp":946684800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>Let <jats:italic>X<\/jats:italic>, <jats:italic>X<\/jats:italic><jats:sup>\u2032<\/jats:sup> be two locally finite, preordered sets and let <jats:italic>R<\/jats:italic> be any indecomposable commutative ring. The incidence algebra <jats:italic>I<\/jats:italic>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>), in a sense, represents <jats:italic>X<\/jats:italic>, because of the well\u2010known result that if the rings <jats:italic>I<\/jats:italic>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) and <jats:italic>I<\/jats:italic>(<jats:italic>X<\/jats:italic><jats:sup>\u2032<\/jats:sup>, <jats:italic>R<\/jats:italic>) are isomorphic, then <jats:italic>X<\/jats:italic> and <jats:italic>X<\/jats:italic><jats:sup>\u2032<\/jats:sup> are isomorphic. In this paper, we consider a preordered set <jats:italic>X<\/jats:italic> that need not be locally finite but has the property that each of its equivalence classes of equivalent elements is finite. Define <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) to be the set of all those functions <jats:italic>f<\/jats:italic> : <jats:italic>X<\/jats:italic> \u00d7 <jats:italic>X<\/jats:italic> \u2192 <jats:italic>R<\/jats:italic> such that <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) = 0, whenever <jats:italic>x<\/jats:italic> \u2a7d <jats:italic>y<\/jats:italic> and the set <jats:italic>S<\/jats:italic><jats:sub><jats:italic>f<\/jats:italic><\/jats:sub> of ordered pairs (<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) with <jats:italic>x<\/jats:italic> &lt; <jats:italic>y<\/jats:italic> and <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) \u2260 0 is finite. For any <jats:italic>f<\/jats:italic>, <jats:italic>g<\/jats:italic> \u2208 <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>), <jats:italic>r<\/jats:italic> \u2208 <jats:italic>R<\/jats:italic>, define <jats:italic>f<\/jats:italic> + <jats:italic>g<\/jats:italic>, <jats:italic>f<\/jats:italic><jats:italic>g<\/jats:italic>, and <jats:italic>r<\/jats:italic><jats:italic>f<\/jats:italic> in <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) such that (<jats:italic>f<\/jats:italic> + <jats:italic>g<\/jats:italic>)(<jats:italic>x<\/jats:italic> + <jats:italic>y<\/jats:italic>) = <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) + <jats:italic>g<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>), <jats:italic>f<\/jats:italic><jats:italic>g<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) = \u2211<jats:sub><jats:italic>x<\/jats:italic>\u2264<jats:italic>z<\/jats:italic>\u2264<jats:italic>y<\/jats:italic><\/jats:sub><jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>z<\/jats:italic>)<jats:italic>g<\/jats:italic>(<jats:italic>z<\/jats:italic>, <jats:italic>y<\/jats:italic>), <jats:italic>r<\/jats:italic><jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>) = <jats:italic>r<\/jats:italic> \u00b7 <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>). This makes <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) an <jats:italic>R<\/jats:italic>\u2010algebra, called the <jats:italic>weak incidence algebra<\/jats:italic> of <jats:italic>X<\/jats:italic> over <jats:italic>R<\/jats:italic>. In the first part of the paper it is shown that indeed <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) represents <jats:italic>X<\/jats:italic>. After this all the essential one\u2010sided ideals of <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) are determined and the maximal right (left) ring of quotients of <jats:italic>I<\/jats:italic><jats:sup>*<\/jats:sup>(<jats:italic>X<\/jats:italic>, <jats:italic>R<\/jats:italic>) is discussed. It is shown that the results proved can give a large class of rings whose maximal right ring of quotients need not be isomorphic to its maximal left ring of quotients.<\/jats:p>","DOI":"10.1155\/s0161171204311130","type":"journal-article","created":{"date-parts":[[2004,10,28]],"date-time":"2004-10-28T11:26:01Z","timestamp":1098962761000},"page":"2835-2845","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Weak incidence algebra and maximal ring of quotients"],"prefix":"10.1155","volume":"2004","author":[{"given":"Surjeet","family":"Singh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fawzi","family":"Al-Thukair","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2004,10,28]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/846181.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/846181.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S0161171204311130","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T23:52:49Z","timestamp":1723074769000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S0161171204311130"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":0,"journal-issue":{"issue":"53","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1155\/S0161171204311130"],"URL":"https:\/\/doi.org\/10.1155\/s0161171204311130","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2004,1]]},"assertion":[{"value":"2003-11-12","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2004-10-28","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}