Abstract
The most natural seminearrings, i.e., the seminearrings of all self-maps of additive semigroups are necessarily multiplicatively regular but they need not be additively regular. The purpose of this paper is to investigate additively regular seminearrings. We mainly focus on the study of congruences in various types of additively regular seminearrings such as additively inverse, additively Clifford and Bandelt seminearrings. We deduce that for a restricted type of additively inverse seminearrings there exists an inclusion preserving bijective correspondence between the set of all normal congruences and that of all full k-ideals. Finally, we characterize those seminearrings which are the subdirect product of a distributive lattice and a zero symmetric near-ring.
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Acknowledgements
The authors are grateful to Prof. M.K. Sen of the University of Calcutta for suggesting the problem and for his constant encouragement and active guidance throughout the preparation of the paper. The authors are also grateful to Prof. L.N. Shevrin and the learned referee for their valuable suggestions and comments to get the previous version of the paper revised into what it is now. We also convey our sincere thanks to the learned referee for his specific suggestion which necessitates the incorporation of Remark 2.9.
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Communicated by Lev N. Shevrin.
Author is grateful to CSIR, Govt. of India, for providing research support as an SRF.
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Sardar, S.K., Mukherjee, R. On additively regular seminearrings. Semigroup Forum 88, 541–554 (2014). https://doi.org/10.1007/s00233-013-9538-z
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DOI: https://doi.org/10.1007/s00233-013-9538-z