Abstract
While establishing the bijection between near-ring congruences and various types of ideals of seminearrings, different kinds of restrictions were imposed either on the seminearrings under consideration or on the near-ring congruences. In this paper we consider a seminearring S without any restriction and establish that there exists an inclusion preserving bijective correspondence between the set \(\{I\subseteq S\) : I is a strong, dense, reflexive and closed additive subsemigroup of S with \(IS\subseteq I\}\) and the set of all near-ring congruences on S. We also show that in any seminearring S, there exists an inclusion preserving bijective correspondence between the set \(\{I\subseteq S\) : I is a strong, dense, reflexive and closed additive subsemigroup of S with \(IS, SI\subseteq I\}\) and the set of all zero-symmetric near-ring congruences on S.
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Acknowledgements
The authors wish to convey their sincere gratefulness to Prof. M. K. Sen of University of Calcutta for his constant inspiration and active guidance throughout the preparation of the paper. The authors are also grateful to the learned referee for many valuable suggestions which has improved the presentation as well as the content of the paper to a great extent. The first author is grateful to CSIR, Govt. of India, for providing research fellowship as an SRF.
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Communicated by László Márki.
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Chakraborty, K., Mukherjee, R. & Sardar, S.K. Near-ring congruences on seminearrings. Semigroup Forum 104, 584–593 (2022). https://doi.org/10.1007/s00233-021-10249-z
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DOI: https://doi.org/10.1007/s00233-021-10249-z