A Note on One Sided and Two Sided PO-Ternary Ideals in PO-Ternary Semiring

Dasari Madhusudhana Rao, Pusapati Siva Prasad, G Srinivasa Rao

Abstract


In this paper the term, left(lateral, right and two sided) PO-ternary ideal, maximal left(lateral, right and two sided) PO-ternary ideal, left (lateral, right and two sided)PO-ternary ideal of T generated by a set A, principal left (lateral, right and two sided)PO-ternary ideal generated by an element a left (lateral, right and two sided) simplePO-ternary semiring are introduced. It is proved that (1) the non-empty intersection of any two left (lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left (lateral, right and two sided) PO-ternary ideal of T. (2) non-empty intersection of any family of left (lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left(lateral, right and two sided) PO-ternary ideal of T. (3) the union of any left PO-ternary ideals of a PO-ternary semiring T is a left PO-ternary ideal of T. (4) the union of any family of left(lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left(lateral, right and two sided) PO-ternary ideal of T. (5) The left (lateral, right and two sided) PO-ternary ideal of a PO-ternary semiring T generated by a non-empty subset A is the intersection of all left(lateral, right and two sided) PO-ternary ideals of T containing A. (6) If T is a PO-ternary semiring and aT then L(a) = (] = (] (M(a) = (] = (], R (a) = (a+ na] = (a?na] and T(a) = (] = (]). (7) A PO-ternary semiring T is a left(lateral, right) simple PO-ternary semiring if and only if (TTa] = T ((TaT ? TTaTT] = T, (aTT] = T) for all aT.


Keywords


PO-ternary Semiring; left PO-ternary ideal; lateral PO-ternary ideal; right PO-ternary ideal; two sided PO-ternary ideal; left simple; lateral simple; right simple.

Full Text:

PDF

References


Chinaram, R., A note on quasi-ideal in semirings, Int. Math. Forum, 3 (2008), 1253-1259.

Dixit, V.N. and Dewan, S., A note on quasi and bi-ideals in ternary semigroups, Int. J. Math. Math. Sci. 18, no. 3 (1995), 501-508.

Dutta, T.K. and Kar, S., On regular ternary semirings, Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientic, New Jersey, 2003, 343{355.

Dutta, T.K. and Kar, S., A note on regular ternary semirings, Kyung-pook Math. J., 46 (2006), 357-365.

Kar, S., On quasi-ideals and bi-ideals in ternary semirings, Int. J. Math. Math. Sc., 18 (2005), 3015-3023.

Lehmer. D. H., A ternary analogue of abelian groups, Amer. J. Math., 59(1932), 329-338.

Lister, W.G., Ternary rings, Trans Amer. Math.Soc., 154 (1971), 37-55.

Madhusudhana Rao. D., Primary Ideals in Quasi-Commutative Ternary Semigroups International Research Journal of Pure Algebra 3(7), 2013, 254-258.

Zhan, J. and Dudek, W.A., Fuzzy hideals of hemirings, Inform. Sci., 177 (2007), 876-886.

Madhusudhana Rao. D. and Srinivasa Rao. G., Structure of Certain Ideals in Ternary Semirings- International Journal of Innovative Science and Modern Engineering (IJISME)ISSN: 2319-6386, Volume- 3 Issue-2, January 2015.

Siva Prasad. P, Madhusudhana Rao. D and Srinivasa Rao. G.,Concepts on PO-Ternary Semirings- International Organization of Scientific Research Journal of Mathematics Volume 11, Issue 3, Ver V, May-Jun 2015, pp 01-06.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Journal of Progressive Research in Mathematics

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright © 2016 Journal of Progressive Research in Mathematics. All rights reserved.

ISSN: 2395-0218.

For any help/support contact us at editorial@scitecresearch.com, jprmeditor@scitecresearch.com.