Abstract
In the endeavour of obtaining semigroup theoretic analogues i.e., the analogues of structure theorems of completely regular semigroups in the setting of additively regular seminearrings we could obtain some results in Mukherjee et al. (Commun Algebra 45(12):5111–5122, 2017). But we could not obtain the analogue of (i) ‘A semigroup is Clifford if and only if it is strong semilattice of groups’ and (ii) ‘ A semigroup is completely regular if and only if it is a union of groups’. In Mukherjee et al. (Commun Algebra, https://doi.org/10.1080/00927872.2018.1524011) we could obtain the analogue of (i) for some restricted type of left (right) Clifford seminearrings. The main purpose of this paper is to complete the remaining task i.e., to obtain the analogue of (ii) for a class of additively completely regular seminearrings. In order to accomplish this we have characterized those seminearrings which are union of near-rings (zero-symmetric near-rings) in the class of additively completely regular seminearrings.
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Notes
Bi-semilattice is a suitable substitute of semilattice in the setting of seminearring.
For semigroup theoretic counterparts of these notations we refer to [4].
\(\checkmark \) denotes the existence and \(\times \) denotes nonexistence. Thus the question asked in the \(2^{nd}\) row is—“Does there exist a seminearring which is RCR as well as GLCR as well as LCR but not GRCR?”
It may be noted that any semiring is also a seminearring.
References
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Acknowledgements
The authors are grateful to Prof. M. K. Sen of University of Calcutta for suggesting the problem. They are also grateful to Prof. S. K. Sardar of Jadavpur University and to Prof. Sen for their active guidance throughout the preparation of the paper. The authors are also grateful to the learned referee for meticulous review and subsequent suggestions for overall improvement of the paper.
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Communicated by Lev N. Shevrin.
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Mukherjee, R., Manna, T. & Pal, P. A note on additively completely regular seminearrings. Semigroup Forum 100, 339–347 (2020). https://doi.org/10.1007/s00233-018-9983-9
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DOI: https://doi.org/10.1007/s00233-018-9983-9