Characteristics of Hyperideals in Ternary Semihyperrings

Authors

DOI:

https://doi.org/10.26713/cma.v13i1.1683

Keywords:

Semihyperring, Ternary semihyperring, Hyperideal, Minimal and maximal hyperideal, (0-)simple ternary semihyperring

Abstract

Ternary semihyperring is an algebraic structure with one binary hyper operation and ternary multiplication. In this paper, we give some properties of hyperideals in ternary semihyperring. We introduce the notion of simple, (0-)simple ternary semihyperring and characterize the minimality and maximality of hyperideals in ternary semihyperring. The relationship between them is investigated in ternary semihyperring extending and generalizing the analogous results for ternary semirings.

Downloads

Download data is not yet available.

References

M. Al-Tahan and B. Davvaz, On some properties of single power cyclic hypergroups and regular relations, Journal of Algebra and Its Applications 16(11) (2017), 1750214, DOI: 10.1142/S0219498817502140.

V. Abramov, R. Kerner and B. Le Roy, Hypersymmetry: A Z3-graded generalization of supersymmetry, Journal of Mathematical Physics 38 (1997), 1650 – 1669, DOI: 10.1063/1.531821.

R. Ameri and H. Hedayati, On k-hyperideals of semihyperrings, Journal of Discrete Mathematical Sciences and Cryptography 10(1) (2007), 41 – 54, DOI: 10.1080/09720529.2007.10698107.

A. Cayley, The Collected Mathematical Papers of Arthur Cayley, Cambridge University Press, Cambridge, URL: https://geographiclib.sourceforge.io/geodesic-papers/cayley-V8.pdf.

S. Chaopraknoi and N. Triphop, Regularity of semihypergroups of infinite matrices, Thai Journal of Mathematics 4(3) (2006), 7 – 11, URL: http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/170.

P. Corsini, Prolegomena of Hypergroup Theory, 2nd edition, Aviani editore, Tricesimo (1993).

P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Advances in Mathematics series (ADMA, Vol. 5), Kluwer Academic Publishers, Dordrecht (2003), DOI: 10.1007/978-1-4757-3714-1.

Y.L. Daletskii and L.A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Letters in Mathematical Physics 39 (1997), 127 – 141, DOI: 10.1023/A:1007316732705.

B. Davvaz, Fuzzy hyperideals in ternary semihyperrings, Iranian Journal of Fuzzy Systems 6(4) (2009), 21 – 36, DOI: 10.22111/ijfs.2009.531.

B. Davvaz, W.A. Dudek and S. Mirvakili, Neutral elements, fundamental relations and n-ary hypersemigroups, International Journal of Algebra and Computation 19(4) (2009), 567 – 583, DOI: 10.1142/S0218196709005226.

B. Davvaz, W.A. Dudek and T. Vougiouklis, A generalization of n-ary algebraic systems, Communications in Algebra 37(4) (2009), 1248 – 1263, DOI: 10.1080/00927870802466835.

H. Qiao, T. Zhao and B. Davvaz, An investigation on S-hypersystems over semihypergroups, Rocky Mountain Journal of Mathematics 51(6) (2021), 2195-2207, DOI: 10.1216/rmj.2021.51.2195.

B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, FL (2007), http://www.santilli-foundation.org/docs/Davvaz.pdf.

V.N. Dixit and S. Dewan, A note on quasi and bi-ideals in ternary semigroups, International Journal of Mathematics and Mathematical Sciences 18 (1995), Article ID 346804, 8 pages, https://doi.org/10.1155/S0161171295000640 DOI: 10.1155/S0161171295000640.

W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift 29 (1929), 1 – 19, https://gdz.sub.uni-goettingen.de/id/PPN266833020_0029?tify={%22pages%22:[5],%22view%22:%22info%22}.

W.A. Dudek, On divisibility in n-semigroups, Demonstratio Mathematica 13 (1980), 355 – 367.

W.A. Dudek and I. Grozdzinska, On ideals in regular n-semigroups, Matematichki Bilten 3-4 (1979-1980), 35 – 44, URL: http://im-pmf.weebly.com/matematicki-bilten---papers-1971-1980.html.

T.K. Dutta and S. Kar, On regular ternary semirings, in: Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific, New Jersey (2003), 343 – 355, DOI: 10.1142/9789812705808_0027.

M. Farshi and B. Davvaz, Relations and homomorphisms of n-hypergroups, European Journal of Combinatorics 44 (2015), 218 – 230, DOI: 10.1016/j.ejc.2014.08.007.

D. Fasino and D. Freni, Existence of proper semihypergroups of type U on the right, Discrete Mathematics 307(2) (2007), 2826 – 2836, DOI: 10.1016/j.disc.2007.03.001.

I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, MA (1994), DOI: 10.1007/978-0-8176-4771-1.

J.W. Grzymala-Busse, Automorphisms of polyadic automata, Journal of the ACM 16(2) (1969), 208 – 219, DOI: 10.1145/321510.321512.

G.A. Miller, Extension of the concept of group of isomorphisms, The American Mathematical Monthly 37(9) (1930), 482 – 484, DOI: 10.1080/00029890.1930.11987117.

R. Kerner, Ternary Algebraic Structures and Their Applications in Physics, Université Pierre-et-Marie-Curie, Paris (2000).

R. Kerner, The cubic chessboard, Classical Quantum Gravity 14(1A) (1997), A203 – A225, URL: https://iopscience.iop.org/article/10.1088/0264-9381/14/1A/017/meta.

C.F. Laywine and G.L. Mullen, Discrete Mathematics Using Latin Squares, 163 pages, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York (1998).

D.H. Lehmer, A ternary analogue of abelian groups, American Journal of Mathematics 54(2) (1932), 329 – 338, DOI: 10.2307/2370997.

J. Łos, J, On the extending of models (I), ´ Fundamenta Mathematicae 42 (1995), 38 – 54, URL: http://eudml.org/doc/213372.

F. Marty, Sur une generalization de la notion de group, In: 8th Congres Math. Scandinaves, Stockholm (1934), pp. 45 – 49.

D. Nikshych and L. Vainerman, Finite Quantum Groupoids and Their Applications, University of California, Los Angeles (2000).

H. Olaru, C. Pelea and I. Purdea, On the completeness of the semihypergroups associated to binaryrelations, Studia Universitatis Babes-Bolyai Mathematica 52(2) (2007), 73 – 85.

A.P. Pojidaev, Enveloping algebras of Fillipov algebras, Communications in Algebra 31(2) (2003), 883 – 900, DOI: 10.1081/AGB-120017349.

F.M. Sioson, Ideal theory in ternary semigroups, Mathematica Japonica 10 (1965), 63 – 64.

Z. Stojakovic and W.A. Dudek, Single identities for varieties equivalent to quadruple systems, Discrete Mathematics 183(1–3) (1998), 277 – 284, DOI: 10.1016/S0012-365X(97)00060-5.

L. Takhtajan, On foundation of the generalized Nambu mechanics, Communications in Mathematical Physics 160 (1994), 295 – 315, DOI: 10.1007/BF02103278.

L. Vainerman and R. Kerner, On special classes of n-algebras, Journal of Mathematical Physics 37(5) (1996), 2553 – 2565, DOI: 10.1063/1.531526.

T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Florida (1994), URL: http://hadronicpress.com/hyperstructures.pdf.

Downloads

Published

23-05-2022
CITATION

How to Cite

Anuradha, T., Prasannam, V. L., & Jayalakshmi, K. (2022). Characteristics of Hyperideals in Ternary Semihyperrings. Communications in Mathematics and Applications, 13(1), 295–306. https://doi.org/10.26713/cma.v13i1.1683

Issue

Vol. 13 No. 1 (2022)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.