{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:15:59Z","timestamp":1761218159670,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T00:00:00Z","timestamp":1562198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincar\u00e9 series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non-   N   -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincar\u00e9 series.<\/jats:p>","DOI":"10.3390\/sym11070881","type":"journal-article","created":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T11:13:18Z","timestamp":1562238798000},"page":"881","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras"],"prefix":"10.3390","volume":"11","author":[{"given":"Oswaldo","family":"Lezama","sequence":"first","affiliation":[{"name":"Seminario de \u00c1lgebra Constructiva-SAC<sup>2<\/sup>, Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Sede Bogot\u00e1 111321, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jaime","family":"Gomez","sequence":"additional","affiliation":[{"name":"Seminario de \u00c1lgebra Constructiva-SAC<sup>2<\/sup>, Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Sede Bogot\u00e1 111321, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,4]]},"reference":[{"key":"ref_1","unstructured":"Ginzburg, V. 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