{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:03:42Z","timestamp":1760241822639,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,19]],"date-time":"2018-09-19T00:00:00Z","timestamp":1537315200000},"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>As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the one-dimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved.<\/jats:p>","DOI":"10.3390\/sym10090417","type":"journal-article","created":{"date-parts":[[2018,9,19]],"date-time":"2018-09-19T10:50:31Z","timestamp":1537354231000},"page":"417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Some Results on Multigranulation Neutrosophic Rough Sets on a Single Domain"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3094-1210","authenticated-orcid":false,"given":"Hu","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Science, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong-Ying","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,19]]},"reference":[{"key":"ref_1","unstructured":"Smarandache, F. 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