{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:53:36Z","timestamp":1760057616508,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,15]],"date-time":"2025-02-15T00:00:00Z","timestamp":1739577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The explicit forms of idempotent and semicentral idempotent triangular matrices over an additively idempotent semiring are obtained. We define a diamond composition of idempotents and give a representation of an idempotent n\u00d7n matrix as an (n\u22121)th degree of a sum of diamond compositions of semicentral idempotents. We construct a decomposition of a strictly upper matrix, a unitriangular matrix, and a nil-clean matrix by semicentral idempotents.<\/jats:p>","DOI":"10.3390\/axioms14020137","type":"journal-article","created":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T03:41:47Z","timestamp":1739763707000},"page":"137","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Idempotent Triangular Matrices over Additively Idempotent Semirings: Decompositions into Products of Semicentral Idempotents"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-0202-5594","authenticated-orcid":false,"given":"Dimitrinka","family":"Vladeva","sequence":"first","affiliation":[{"name":"Department of Algebra and Logic, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bistarelli, S. 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