Table of Contents
Retracted

This article has been retracted as it is essentially identical in title and technical content with the published article “Idempotent Elements of the Endomorphism Semiring of a Finite Chain,” by Ivan Trendafilov and Dimitrinka Vladeva, published in Comptes Rendus de L'Académie Bulgare des Sciences 66 (2013), no. 5, 621-628.

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References

  1. I. Trendafilov and D. Vladeva, “Idempotent elements of the endomorphism semiring of a finite chain,” ISRN Algebra, vol. 2013, Article ID 120231, 9 pages, 2013.
ISRN Algebra
Volume 2013, Article ID 120231, 9 pages
http://dx.doi.org/10.1155/2013/120231
Research Article

Idempotent Elements of the Endomorphism Semiring of a Finite Chain

1Department of Algebra and Geometry, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, Bulgaria
2Department of Mathematics and Physics, University of Forestry, Sofia, Bulgaria

Received 30 November 2012; Accepted 19 December 2012

Academic Editors: S. Dascalescu, A. Vourdas, and M. Walton

Copyright © 2013 Ivan Trendafilov and Dimitrinka Vladeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well-known semiring. The construction of an equivalence relation such that any equivalence class contains just one idempotent is proposed. In our main result we prove that such an equivalence class is a semiring and find its order. We prove that the set of all idempotents with certain jump points is a semiring.