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The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B\u2010module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi\u2010poset version of the completion of Dedekind and\u00a0MacNeille.<\/jats:p>","DOI":"10.1002\/malq.202100029","type":"journal-article","created":{"date-parts":[[2022,2,16]],"date-time":"2022-02-16T03:57:48Z","timestamp":1644983868000},"page":"159-170","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantum B\u2010modules"],"prefix":"10.1002","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7487-0840","authenticated-orcid":false,"given":"Xia","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences South China Normal University 510631 Guangzhou China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wolfgang","family":"Rump","sequence":"additional","affiliation":[{"name":"Institute for Algebra and Number Theory University of Stuttgart Pfaffenwaldring 57, 70550 Stuttgart Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,2,15]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129500000189"},{"volume-title":"Abstract and Concrete Categories: The Joy of Cats","year":"1990","author":"Ad\u00e1mek J.","key":"e_1_2_7_3_1"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0633-1"},{"volume-title":"Algebraic approach to tense operators, Research and Exposition in Mathematics","year":"2015","author":"Chajda I.","key":"e_1_2_7_5_1"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90045-4"},{"volume-title":"An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society","year":"1984","author":"Joyal A.","key":"e_1_2_7_7_1"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/S1570-7954(07)05006-1"},{"key":"e_1_2_7_9_1","unstructured":"C. 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