{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:35:42Z","timestamp":1760150142252,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,30]],"date-time":"2023-09-30T00:00:00Z","timestamp":1696032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["11501331"],"award-info":[{"award-number":["11501331"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper introduces a representation of subnetworks of a network \u0393 consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network \u0393 form a quasi-semilattice L(\u0393), namely a network quasi-semilattice.Two equivalences \u03c3 and \u03b4 are defined on L(\u0393). Each \u03b4 class forms a semilattice and also has an order structure with the maximum element and minimum elements. Here, the minimum elements correspond to spanning trees in graph theory. Finally, we show how graph inverse semigroups, Leavitt path algebras and Cuntz\u2013Krieger graph C*-algebras are constructed in terms of relations.<\/jats:p>","DOI":"10.3390\/axioms12100943","type":"journal-article","created":{"date-parts":[[2023,10,2]],"date-time":"2023-10-02T04:39:30Z","timestamp":1696221570000},"page":"943","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Quasi-Semilattices on Networks"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1203-4641","authenticated-orcid":false,"given":"Yanhui","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2810-8999","authenticated-orcid":false,"given":"Dazhi","family":"Meng","sequence":"additional","affiliation":[{"name":"College of Applied Science, Beijing University of Technology, 100 Pingleyuan, Beijing 100124, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2246","DOI":"10.1126\/science.1103330","article-title":"Use of Logic Relationships to Decipher Protein Network Organization","volume":"306","author":"Bowers","year":"2004","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.physrep.2013.08.002","article-title":"Clustering and community detection in directed networks: A survey","volume":"533","author":"Malliaros","year":"2013","journal-title":"Phys. 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