{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:12:12Z","timestamp":1760145132794,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,19]],"date-time":"2024-06-19T00:00:00Z","timestamp":1718755200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.","award":["PNURSP2024R231"],"award-info":[{"award-number":["PNURSP2024R231"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of idempotent transformations with restricted range, revealing that these automorphisms are inner ones and induced by the units of S. Additionally, we establish that the automorphism group Aut(S) is isomorphic to US, the group of units of S. These findings extend previous results on semigroups of transformations, enhancing their applicability and providing a more unified theory. The practical implications of this work span multiple fields, including automata theory, coding theory, cryptography, and graph theory, offering tools for more efficient algorithms and models. By simplifying complex concepts and providing a solid foundation for future research, this study makes significant contributions to both theoretical and applied mathematics.<\/jats:p>","DOI":"10.3390\/sym16060769","type":"journal-article","created":{"date-parts":[[2024,6,20]],"date-time":"2024-06-20T05:27:19Z","timestamp":1718861239000},"page":"769","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Centralizers of Idempotents with Restricted Range"],"prefix":"10.3390","volume":"16","author":[{"given":"Dilawar J.","family":"Mir","sequence":"first","affiliation":[{"name":"Department of Mathematics, CDOE, Chandigarh University, Mohali 140413, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7856-2861","authenticated-orcid":false,"given":"Amal S.","family":"Alali","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,19]]},"reference":[{"key":"ref_1","first-page":"572","article-title":"The Algebraic Theory of Semigroups","volume":"s1-44","author":"Clifford","year":"1961","journal-title":"Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"163","DOI":"10.2307\/1969317","article-title":"On the Structure of Semigroups","volume":"54","author":"Green","year":"1951","journal-title":"Ann. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1017\/S0305004100031935","article-title":"On Semigroups of Matrices","volume":"53","author":"Munn","year":"1957","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_4","unstructured":"Petrich, M. (1984). Inverse Semigroups, Wiley."},{"key":"ref_5","unstructured":"Howie, J.M. (1995). Fundamentals of Semigroup Theory, Oxford University Press. 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Vyss Ucebn Zaved Mat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"73","DOI":"10.4153\/CMB-1966-009-7","article-title":"Automorphisms of semigroup of all relations on a set","volume":"9","author":"Magill","year":"1966","journal-title":"Canas. Math. Bull."},{"key":"ref_11","first-page":"531","article-title":"On symmetric generalised groups","volume":"33","author":"Liber","year":"1953","journal-title":"Math. Sbornik N. S."},{"key":"ref_12","first-page":"105","article-title":"Semigroups and rings of endomorphisms of linear spaces-I","volume":"45","author":"Gluskin","year":"1965","journal-title":"Am. Math. Soc. Transl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1017\/S144678870002396X","article-title":"Automorphisms of Transformation semigroups","volume":"20","author":"Sullivan","year":"1975","journal-title":"J. Austral Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BF02573654","article-title":"On Automorphisms of Transformation semigroups","volume":"48","author":"Levi","year":"1994","journal-title":"Semigroup Forum"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1017\/S001309150002263X","article-title":"Automorphisms of normal Transformation semigroups","volume":"2","author":"Levi","year":"1985","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_16","first-page":"2250032","article-title":"On Automorphisms of Monotone Transformation posemigroups, Asian-Eur","volume":"15","author":"Mir","year":"2022","journal-title":"J. Math"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Shah, A.H., Mir, D.J., and Quinn-Gregson, T. (2024). Inner Automorphisms of Clifford Monoids. Hacettepe J. Math. Stat., 1\u201315.","DOI":"10.15672\/hujms.1244782"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1017\/S1446788700034455","article-title":"Some results concerning a transformation semigroups","volume":"19","author":"Symons","year":"1975","journal-title":"J. Austral Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1017\/S1446788700016232","article-title":"Normal transformation semigroups","volume":"22","author":"Symons","year":"1976","journal-title":"J. Austral Math. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/S0021-8693(03)00499-X","article-title":"Automorphisms of Centralizers of idempotents","volume":"269","author":"Araujo","year":"2003","journal-title":"J. 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