The structure of certain unique classes of seminearrings

Year 2024, , 139 - 148, 09.01.2024

G. Manikandan Perumal Ramachandran , P. Madhusoodhanan

https://doi.org/10.24330/ieja.1402798

Abstract

In this paper, we introduce the classes of $\alpha$ and strictly-$\alpha$ seminearrings and establishes some of their properties, mostly in relation to the possession of a mate
function. Then we get the criterion for an $\alpha$-seminearring to become a strictly-$\alpha$ seminearring. We also obtain a complete characterisations of $\alpha$ and strictly-$\alpha$ seminearrings and proved certain results for $\alpha$ and strictly-$\alpha$ seminearrings via certain unique classes of seminearrings.

Keywords

$\alpha$-seminearring, strictly-$\alpha$ seminearring, distributively generated seminearring, mate function, ideal

References

• J. Ahsan, Seminear-rings characterized by their $\mathcal{S}$-ideals, I, Proc. Japan Acad. Ser. A Math. Sci., 71(5) (1995), 101-103.
• J. Ahsan, Seminear-rings characterized by their $\mathcal{S}$-ideals, II, Proc. Japan Acad. Ser. A Math. Sci., 71(6) (1995), 111-113.
• R. Balakrishnan and R. Perumal, Left duo seminear-rings, Scientia Magna, 8(3) (2012), 115-120.
• R. Balakrishnan and S. Suryanarayanan, $P(r,m)$ near-rings, Bull. Malays. Math. Sci. Soc., 23(2) (2000), 117-130.
• R. Balakrishnan and S. Suryanarayanan, On $P_k$ and $P_k'$ near-rings, Bull. Malays. Math. Sci. Soc., 23(1) (2000), 9-24.
• T. Boykett, Seminearring models of reversible computation, Univ., Institut fur Mathematik, 1997, 19 pp.
• G. Gratzer, Universal Algebra, Springer, New York, 2008.
• H. Hoogewijs, Semi-nearring embeddings, Med. Konink. Vlaamse Acad. Wetensch. Lett. Schone Kunst, Belgie Kl. Wetensch,32(2) (1970), 11 pp.
• W. G. van Hoorn and B. van Rootselaar, Fundamental notions in the theory of seminearrings, Compositio Math., 18 (1967), 65-78.
• F. Hussain, M. Tahir, S. Abdullah and N. Sadiq, Quotient seminear-rings, Indian Journal of Science and Technology, 9(38) (2016), 1-7.
• S. A. Huq, Embedding problems, module theory and semisimplicity of seminearrings, Ann. Soc. Sci. Bruxelles Sér. I., 103(1) (1989), 49-62.
• G. Manikandan and R. Perumal, Mate and mutual mate functions in a seminearring, AIMS Math., 5(5) (2020), 4974-4982.
• G. Manikandan and R. Perumal, Mid-units in duo seminearrings, J. Phys. Conf. Ser., 1850 (2021), 012037 (5 pp).
• G. Manikandan and R. Perumal, A note on left duo seminearrings, Palest. J. Math., 11(Special Issue I) (2022), 24-28.
• G. Manikandan, R. Perumal and R. Arulprakasam, Strong K(r,s) seminear-rings, AIP Conf. Proc., 2112(1) (2019), 020116 (5 pp).
• R. Perumal and R. Balakrishnan, Ideals in $P_k$ and $P_k'$ seminear-rings, Malaya J. Mat., S(1) (2013), 66-70.
• G. Pilz, Near-Rings: The Theory and its Applications, Mathematics Studies, 23, North-Holland Publishing Co., Amsterdam, 1977.
• B. van Rootselaar, Algebraische kennzeichnung freier wortarithmetiken, Compositio Math., 15 (1963), 156-168.
• M. Shabir and I. Ahmed, Weakly regular seminearrings, Int. Electron. J. Algebra, 2 (2007), 114-126.
• H. J. Weinert, Seminearrings, seminearfields and their semigroup-theoretical background, Semigroup Forum, 24(2-3) (1982), 231-254.
• H. J. Weinert, Partially and fully ordered seminearrings and nearrings, Near-Rings and Near-Fields, North-Holland Math. Stud., 137 (1987), 277-294.
• H. J. Weinert, Extensions of seminearrings by semigroups of right quotients, Recent developments in the algebraic, analytical and topological theory of semigroups (Oberwolfach, 1981), 412–486, Lecture Notes in Math., Springer-Verlag, Berlin, 998 (1983).