On additively completely regular seminearrings: II

Abstract

As a continuation of our previous work, an attempt has been made to obtain some sort of analog of structure theorem of Clifford semigroups in the setting of seminearrings. To accomplish this, the notion of strong bi-semilattice of seminearrings has been introduced. Then those left (right) Clifford seminearrings, which are strong bi-semilattice of near-rings (zero-symmetric near-rings) and strong distributive lattice of near-rings (zero-symmetric near-rings) have been characterized.

Keywords:

• E⁺-unitary seminearring
• strong bi-semilattice of (zero-symmetric) near-rings
• strong distributive lattice of (zero-symmetric) near-rings

2010 MATHEMATICS Subject Classification:

• 16Y30
• 16Y60
• 16Y99

Acknowledgements

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 article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Bi-semilattice is a suitable substitute of semilattice in the setting of seminearring.

2 For semigroup theoretic counterparts of these notations we refer to Ref. [7].

3 An element d is said to be a distributive element if $d·(a+b)=d·a+d·b$ for all a, $b\in S$.

3 It may be noted that any semiring is also a seminearring.