Study of Properties of Several Kinds of Hyperideals in Ordered Ternary Semihypergroups

Thawhat Changphas, Bijan Davvaz


A non-empty subset S together with an associative function f from S × S×S into the family of all non-empty subsets of S is called a ternary semihypergroup. In this paper, we consider a semihypergroup (S, f) besides a binary relation ≤, where ≤ is a partial order relation on S such that satisfies the monotone condition. This structure is called an ordered ternary semihypergroup. We introduce and investigate the notions of bi-hyperideal and quasi-hyperideal in ordered ternary semihyperroups. In particular, we prove that an ordered ternary semihypergroup is left and right simple if and only if it does not contain proper bi-hyperideals


Algebraic hyperstructure; ordered ternary semihypergrouip; ternary subsemihypergroup; (bi-, quasi-) hyperideal; prime hyperideal; quasi-simple.

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Anvariyeh, S.M., Mirvakili, S. and Davvaz, B., Combinatorial aspects of n-ary polygroups and n-ary color schemes, European Journal of Combinatorics, 34 (2013), 207-216.

Bonansinga, P. and Corsini, P., On semihypergroup and hypergroup homomorphisms, Boll. Un. Mat. Ital. B (6) 1(2) (1982), 717-727.

Chvalina, J., Commutative hypergroups in the sense of Marty and ordered sets, General algebra and ordered sets (Horn Lipova, 1994), 19-30.

Davvaz, B., Some results on congruences in semihypergroups, Bull. Malays. Math. Soc. (2), 23 (2000), 53-58.

Davvaz, B., Characterizations of sub-semihypergroups by various triangular norms, Czechoslovak Mathematical Journal, 55(4) (2005), 923-932.

Davvaz, B. and Leoreanu-Fotea, V., Binary relations on ternary semihypergroups, Communications in Algebra, 38(10) (2010) 3621-3636.

Davvaz, B. and Vougiouklis, T., n-Ary hypergroups, Iran. J. Sci. Technol. Trans. A, 30(2) (2006), 165-174.

De Salvo, M., Freni, D. and Lo Faro, G., Fully simple semihypergroups, J. Algebra, 399 (2014), 358-377.

Dornte, W., Untersuchungen ber einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift, 29(1) (1929) 1-19.

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

Freni, D., Minimal order semihypergroups of type U on the right, II, J. Algebra, 340 (2011), 77-89.

Heidari, D. and Davvaz, B., On ordered hyperstructures, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 73(2) (2011), 85-96.

Hila, K., Naka, K., Leoreanu-Fotea, V. and Sadiku, S., Algebraic hyperstructures of soft sets associated with ternary semihypergroups, Ital. J. Pure Appl. Math., 30 (2013), 349-372.

Hila, K., Davvaz, B. and Naka, K., On Quasi-hyperideals in semihypergroups, Communications in Algebra, 39 (2011), 4183-4194.

Hila, K., Davvaz, B. and Dine, J., Study on the structure of Γ-semihypergroups, Communications in Algebra, 40(8) (2012), 2932-2948.

Iampan, A., On ordered ideal extensions of ordered ternary semigroups, Lobachevskii J. Math., 31(1) (2010), 13-17.

Marty, F., Sur une generalization de la notion de groupe, 8

iem Congres Math. Scandinaves, Stockholm, (1934) 45-49.

Mirvakili, S. and Davvaz, B., Applications of strongly transitive geometric spaces to n-ary hypergroups, ARS Combinatoria, 109 (2013), 193-227.

Mirvakili, S. and Davvaz, B., On some combinatorial aspects of transposition n-ary hypergroups, Carpathian Journal of

Mathematic, 30(1) (2014), 109-116.

Naka, K. and Hila, K., Some properties of hyperideals in ternary semihypergroups, Math. Slovaca, 63(3) (2013), 449-468.

Lehmer, D.H., A ternary analogue of abelian groups, American Journal of Mathematics, 54 (1932), 329-338.

Leoreanu, V., About the simplifiable cyclic semihypergroups, Ital. J. Pure Appl. Math., 7 (2000), 69-76.

Leoreanu-Fotea, V. and Davvaz, B., n-hypergroups and binary relations, European Journal of Combinatorics, 29 1207-1218.

Los, J., On the extending of models I, Fundamenta Mathematicae, 42 (1955), 38-54.

M.L. Santiago, M.L. and Bala, S.S., Ternary semigroups, Semigroup Forum, 81 (2010) 380-388.

Sioson, F.M., Ideal theory in ternary semigroups, Mathematica Japonica, 10 (1965) 63-84



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