Abstract: In this paper, we introduce the notion of complementation of a Γ-semiring. By using the concepts of centreless, simple, additive idempotent and multiplicative Γ-idempotent Γ-semiring, we characterize some results of semirings to Γ-semirings. Further, we prove that if R is a centreless Γ-semiring with strong identity, then the set of all complemented elements of R forms a strong Γ-idempotent, commutative, simple Γ-semiring. |
Keywords and phrases: additive idempotent and strong multiplicative Γ-idempotent Γ-semiring, complemented elements in a Γ-semiring.
Received: February 18, 2021; Revised: March 22, 2021; Accepted: March 23, 2021; Published: April 28, 2021
How to cite this article: Tilak Raj Sharma and Shweta Gupta, Complementation of a gamma semiring, JP Journal of Algebra, Number Theory and Applications 50(2) (2021), 101-112. DOI: 10.17654/NT050010101
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] J. S. Golan, Semirings and their Applications, Kluwer Academic Publishers, 1999. [2] J. S. Golan, The theory of semirings with applications in mathematics and theoretical computer science, Pitman Monograph and Surveys in Pure and Applied Mathematics, 1992. [3] H. Hedayati and K. P. Shum, An Introduction to -semirings, International Journal of Algebra 5(15) (2011), 709-726. [4] Peter T. John Stone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [5] Ernest G. Manes and Michael A. Arbib, Algebraic Approaches to Program Semantics, Springer-Verlag, Berlin, 1986. [6] M. M. Krishana Rao, Gamma semiring-I, Southeast Asian Bull. Math. 19 (1995), 49-54. [7] M. M. K. Rao, On field gamma semiring and completed gamma semiring with identity, Bulletin of the International Mathematical Virtual Institute 8 (2018), 189-202. [8] S. K. Sardar and B. C. Saha, On Nobusawa gamma semirings, Universitatee Din Bacau Studii Si Cercetari Stiinifice Seria: Mathematica 18 (2008), 283-306. [9] Tilak Raj Sharma and Shweta Gupta, Some conditions on Γ-semirings, JCISS 41(1-2) (2016), 79-87. [10] H. S. Vandiver, Note on a simple type of algebra in which cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40 (1934), 187-191.
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