{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T17:57:50Z","timestamp":1754157470807,"version":"3.41.2"},"reference-count":16,"publisher":"Emerald","issue":"3\/4","license":[{"start":{"date-parts":[[2009,4,10]],"date-time":"2009-04-10T00:00:00Z","timestamp":1239321600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009,4,10]]},"abstract":"<jats:sec><jats:title content-type=\"abstract-heading\">Purpose<\/jats:title><jats:p>The purpose of this paper is to study free <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010module, using the language of categories and functors for the general description of <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules generated by <jats:italic>L<\/jats:italic>\u2010fuzzy set. In the language of categories and functors, an <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules generated by <jats:italic>L<\/jats:italic>\u2010fuzzy set is called a free object in the category of <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules determined by <jats:italic>L<\/jats:italic>\u2010fuzzy set.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Design\/methodology\/approach<\/jats:title><jats:p>Category theory is used to study the existent quality, unique quality and material structure of <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules generated by <jats:italic>L<\/jats:italic>\u2010fuzzy set.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Findings<\/jats:title><jats:p>The paper gives the uniqueness, structure and existence theorems of free object in the category of <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules determined by <jats:italic>L<\/jats:italic>\u2010fuzzy set, and the authors prove that the fuzzy free functor is left adjoint to the fuzzy underlying functor.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Research limitations\/implications<\/jats:title><jats:p>Some property of free <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules will need to be further researched.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Originality\/value<\/jats:title><jats:p>The paper defines a new class of <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules, i.e. free <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules, research and explore free <jats:italic>L<\/jats:italic>\u2010fuzzy left R\u2010modules in theory.<\/jats:p><\/jats:sec>","DOI":"10.1108\/03684920910944227","type":"journal-article","created":{"date-parts":[[2009,5,30]],"date-time":"2009-05-30T07:00:28Z","timestamp":1243666828000},"page":"506-512","source":"Crossref","is-referenced-by-count":2,"title":["Free object in the category of <i>L<\/i>\u2010fuzzy left R\u2010modules determined by <i>L<\/i>\u2010fuzzy set"],"prefix":"10.1108","volume":"38","author":[{"given":"Jian\u2010Gang","family":"Tang","sequence":"first","affiliation":[]},{"given":"Mao\u2010Kang","family":"Luo","sequence":"additional","affiliation":[]},{"given":"Miao","family":"Liu","sequence":"additional","affiliation":[]}],"member":"140","reference":[{"key":"key2022012520200335500_b6","doi-asserted-by":"crossref","unstructured":"Goguen, J.A. 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