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On additively commutative near-ring congruences on seminearrings

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Abstract

In the class of natural seminearrings i.e., the seminearrings of all self-maps of additive semigroups, the additively regular seminearrings are precisely the seminearrings of self-maps of additively regular semigroups. Hence there are plenty of natural seminearrings which are not additively regular. In this paper we consider a restricted type of such seminearrings and characterize their additively commutative near-ring congruences in terms of normal subseminearrings. In this class of seminearrings, we also obtain an isomorphism between the lattice of all additively commutative near-ring congruences and the lattice of all normal subseminearrings. This isomorphism is used to show that the set of all additively commutative near-ring congruences of the class of seminearrings under consideration forms a distributive lattice.

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Acknowledgments

The authors are grateful to Prof. M. K. Sen of University of Calcutta for suggesting the problem and for constant encouragement and active guidance throughout the preparation of the paper. The authors also convey their sincere thanks to Prof. L. N. Shevrin and the learned referee.

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    Rajlaxmi Mukherjee (Pal), Sujit Kumar Sardar & Pavel Pal

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  1. Rajlaxmi Mukherjee (Pal)
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  2. Sujit Kumar Sardar
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  3. Pavel Pal
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Corresponding author

Correspondence to Sujit Kumar Sardar.

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Communicated by Lev Shevrin.

The first author is grateful to CSIR, Govt. of India, for providing research support as an SRF.

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Mukherjee (Pal), R., Sardar, S.K. & Pal, P. On additively commutative near-ring congruences on seminearrings. Semigroup Forum 91, 573–583 (2015). https://doi.org/10.1007/s00233-014-9664-2

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  • DOI: https://doi.org/10.1007/s00233-014-9664-2

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