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Complemented and completely regular $\Gamma-$ semirings | ||
| Journal of Algebra and Related Topics | ||
| دوره 14، شماره 1، آبان 2026، صفحه 115-126 اصل مقاله (151.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.29305.1747 | ||
| نویسندگان | ||
| T. R. Sharma* 1؛ R. Kumar2 | ||
| 1Himachal Pradesh University Regional Centre, Dharamshala (H.P.), India | ||
| 2Government Degree College, Dharamshala(H.P.), India | ||
| چکیده | ||
| This paper investigates the additive and multiplicative properties of complemented and completely regular $\Gamma-$ semirings. We prove many results on different structures of $\Gamma-$ semirings like anti-inverse, quasi-separative, distributive, and partial order. Boolean $\Gamma-$ semiring is demonstrated by applying the concept of a completely regular $\Gamma-$ semiring. Finally, by using the idea of simple and completely regular $\Gamma-$ semiring, we define a relation $\leq$ on $R$ such that $x \leq y$ if and only if $x+y+1 = x\alpha y$ for all $x,y \in R, \alpha \in \Gamma $ and prove that $R$ is a partially ordered $\Gamma-$ semiring. | ||
| کلیدواژهها | ||
| Regular band؛ $\Gamma-$semigroup؛ Quasi-separative؛ Partially ordered $\Gamma-$ semiring | ||
| مراجع | ||
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[1] H. K. Chandrakala and A. Rajeshwari, Completely regular semiring, International Journal of Science and Research, (4) 9 (2020), 1696-1699. [2] T. K. Dutta and S. K. Sardar, Semiprime ideals and irreducible ideals of a Γ−semiring, Novi Sad Journal of Mathematics, (1) 30 (2000), 97-108. [3] T. K. Dutta and S. K. Sardar, Study of notherian Γ− semiring via operator semirings, Southeast Asian Bulletin of Mathematics, (4) 25 (2002), 599-608. [4] T. K. Dutta and S. K. Sardar, On the operator semirings of a Γ− semiring, Southeast Asian Bulletin of Mathematics, (2) 26 (2003), 203-213. [5] G. Mallikarjuna and G. Shobhalatha, Properties of complemented semirings, International Organization of Scientific Research, Journal of Mathematics, (5) 12 (2016), 107-112. [6] N. Nobusawa, On a generalization of the ring theory, Osaka Journal of Mathematics, 1 (1964), 81-89. [7] M. M. K. Rao, Γ− semirings-I, South East Asian Bulletin of Mathematics, (1) 19 (1995), 49-54. [8] M. M. K. Rao, On field Γ− semiring and complemented Γ− semiring with identity, Bulletin of International Mathematical Virtual Institute, 8 (2017), 85-99. [9] M. K. Sen, Γ− semigroup, Proceedings of International Conference of Algebra and its Applications, Decker Publication, New York, (1981), 301-308. [10] M. K. Sen, and N. K. Saha, On Γ− semigroup, Bulletin of Calcutta Mathematical Society, (1986), 180-186. [11] N. V. Subrahmanyam, Boolean semirings, Mathematische Annalen, 148 (1962), 395-401. [12] T. R. Sharma, Lattices in Γ− semirings, International Journal of Creative Research Thoughts, (11) 8 (2020), 3360-3366. [13] T. R. Sharma and S. Gupta, Some conditions on Γ− semirings, Journal of Combinatorics, Information and System Sciences, (1-3) 41 (2016), 79-87. [14] T. R. Sharma and S. Gupta, Ideals in Γ− semirings, Journal of Combinatorics, Information and System Sciences, (1-4) 42 (2017), 165-177. [15] T. R. Sharma and S. Gupta, Ideals of a bourne factor Γ− semirings, Proceedings of National Conference on Advances in Mathematical Sciences, Research Journal of Science and Technology, (1) 9 (2017), 171-174. [16] T. R. Sharma and S. Gupta, Complementation of Γ− semirings, JP Journal of Algebra, Number theory and Applications, 50 (2021), 101-112. [17] T. R. Sharma and H. K. Ranote, Some properties of a Γ− semiring, JP Journal of Algebra, Number theory and Applications, (2) 52 (2021), 163-177. | ||
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