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Near-ring congruences on additively regular seminearrings

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Abstract

In this paper we first obtain analogues of some results of LaTorre (Semigroup Forum 24(1):327–340, 1982) in the setting of additively regular seminearrings which in turn not only give rise to refinements of some important results viz. Propositions 3.16, 3.17, Theorem 3.20 of Sardar and Mukherjee (Semigroup Forum 93(3):629–631, 2016) and Theorem 3.22 of Sardar and Mukherjee (Semigroup Forum 88(3):541–554, 2014) (involving mainly near-ring congruences i.e., normal congruences and normal full k-ideals of additively inverse seminearrings) but also answer partially a question raised in Sardar and Mukherjee (Semigroup Forum 88(3):541–554, 2014). Finally we study the lattice structures of near-ring congruences and normal full k-ideals in distributively generated additively regular seminearrings.

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Notes

  1. For an additively regular seminearring near-ring congruences and normal congruences coincide (cf. Theorem 1) and so they are used here interchangeably.

  2. The analogue of the second part is proved in Theorem 8.

  3. For definition of Hw we refer to Definition 5.

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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 meticulous perusal of the paper and subsequent valuable suggestions. The first author is grateful to CSIR, Govt. of India, for providing research fellowship as an SRF.

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Authors and Affiliations

  1. Jadavpur University, Jadavpur, Kolkata, 700032, India

    Kamalika Chakraborty & Sujit Kumar Sardar

  2. Bankura University, Bankura, 722155, India

    Pavel Pal

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  1. Kamalika Chakraborty
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  2. Pavel Pal
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  3. Sujit Kumar Sardar
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Correspondence to Sujit Kumar Sardar.

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Communicated by Lev Shevrin.

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Chakraborty, K., Pal, P. & Sardar, S.K. Near-ring congruences on additively regular seminearrings. Semigroup Forum 101, 285–302 (2020). https://doi.org/10.1007/s00233-020-10123-4

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