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Keywords:
additive inverse semirings; regular semirings; orthodox semirings
additive inverse semirings; regular semirings; orthodox semirings
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.
References:
[1] Chaptal N.: Anneaux dont le demi groupe multiplicatif est inverse. C. R. Acad. Sci. Paris, Ser. A-B, 262 (1966), 247–277. MR 0190177 | Zbl 0133.29001
[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
[3] Howie J. M., Introduction to the theory of semigroups. : Academic Press. 1976.
[4] Karvellas P. H.: Inverse semirings. J. Austral. Math. Soc. 18 (1974), 277–288. MR 0366991
[5] Zeleznekow J.: Regular semirings. Semigroup Forum 23 (1981), 119–136. MR 0641993
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