On additively completely regular seminearrings

ABSTRACT

In an attempt to investigate the situation arising out of replacing additive regularity by additive complete regularity in our previous study on additively regular seminearrings, we introduce the notions of left (right) completely regular seminearrings and characterize left (right) completely regular seminearrings as bi-semilattices of left (resp., right) completely simple seminearrings. We also define left (right) Clifford seminearrings and show that they are precisely bi-semilattices of near-rings (resp., zero-symmetric near-rings).

KEYWORDS:

• Additively completely regular seminearring
• Green’s relation
• left (right) Clifford seminearring
• left (right) completely regular seminearring
• left (right) completely simple seminearring

2010 AMS SUBJECT CLASSIFICATION:

• 16Y30
• 16Y60
• 16Y99
• 06B10

Acknowledgment

The authors are grateful to Professor 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 are also grateful to the learned referee for his meticulous review and subsequent suggestions for overall improvement, particularly the presentation of the paper.

Notes

¹Throughout this paper ‘bi-semilattice’ stands for ‘meet distributive bi-semilattice’ [8].

²The semigroup direct product of any non-trivial group (G,+) and any non-trivial semilattice (L,+) is such a semigroup.

³In fact (S^′,+) is the semigroup S⁰ where (S,+) is the semigroup {1,−1} with x+y = |x|y as the composition.

⁴The set of all non-zero real numbers, with a+b: = |a|b, forms such a semigroup.

⁵In view of Proposition 3.5, direct product of a left Clifford and a right Clifford seminearring is a left Clifford seminearring.