# Article

 Title: A note on orthodox additive inverse semirings (English) Author: Sen, M. K. Author: Maity, S. K. Language: English Journal: Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica ISSN: 0231-9721 Volume: 43 Issue: 1 Year: 2004 Pages: 149-154 Summary lang: English . Category: math . Summary: We show in an additive inverse regular semiring $(S, +, \cdot )$ with $E^{\bullet }(S)$ as the set of all multiplicative idempotents and $E^+(S)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e, f \in E^{\bullet }(S)$, $ef \in E^+(S)$ implies $fe\in E^+(S)$. (ii) $(S, \cdot )$ is orthodox. (iii) $(S, \cdot )$ is a semilattice of groups. This result generalizes the corresponding result of regular ring. (English) Keyword: additive inverse semirings Keyword: regular semirings Keyword: orthodox semirings MSC: 16A78 MSC: 16E50 MSC: 16Y60 MSC: 20M07 MSC: 20M10 idZBL: Zbl 1067.16070 idMR: MR2124613 . Date available: 2009-08-21T12:55:15Z Last updated: 2012-05-04 Stable URL: http://hdl.handle.net/10338.dmlcz/132936 . Reference: [1] Chaptal N.: Anneaux dont le demi groupe multiplicatif est inverse.C. R. Acad. Sci. Paris, Ser. A-B, 262 (1966), 247–277. Zbl 0133.29001, MR 0190177 Reference: [2] Golan J. S.: The Theory of Semirings with Applications in Mathematics, Theoretical Computer Science. : Pitman Monographs and Surveys in Pure and Applied Mathematics 54, Longman Scientific., 1992. MR 1163371 Reference: [3] Howie J. M., Introduction to the theory of semigroups. : Academic Press., 1976. Reference: [4] Karvellas P. H.: Inverse semirings.J. Austral. Math. Soc. 18 (1974), 277–288. MR 0366991 Reference: [5] Zeleznekow J.: Regular semirings.Semigroup Forum 23 (1981), 119–136. MR 0641993 .

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