Niovi KEHAYOPULU

On the paper “A study on (strong) order-congruences in ordered semihypergroups”

Throughout the paper in the title by Jian Tang, Yanfeng Luo and Xiangyun Xie in Turk J Math 42 (2018) the following lemma has been used. Lemma: Let (S, ∗) be a semihypergroup and ρ an equivalence relation on S. Then (i) If ρ is a congruence, then (S/ρ, ⊗) is a semihypergroup with respect to the hyperoperation (a)ρ ⊗ (b)ρ = ∪ c∈a∗b (c)ρ . (ii) If ρ is a strong congruence, then (S/ρ, ⊗) is a semigroup with respect to the operation (a)ρ ⊗ (b)ρ = (c)ρ for all c ∈ a ∗ b . The property (i) of the paper is certainly wrong as ∪ c∈a∗b (c)ρ is a subset of S and not a nonempty subset of S/ρ as it should be. Property (ii) has no sense in the way is written. In addition, according to the authors, as an application of the results of this paper they solved the open problem on ordered semihypergroups given by Davvaz, Corsini and Changphas in European J Combin 44 (2015). The problem is that the above mentioned problem has not been solved in the above mentioned article; we point out the reason, and we solve it in the present paper. Some further related results; also results necessarily for the completeness of the paper are given. Examples illustrate the results.

PDF

___

[1] Clifford AH, Preston GB. The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7 American Mathematical Society, Providence, R.I. 1961; xv+224 pp.
[2] Davvaz B, Corsini P, Changphas T. Relationship between ordered semihypergroups and ordered semigroups by using pseudorder. European Journal of Combinatorics 2015; 44: part B, 208–217. doi:10.1016/j.ejc.2014.08.006
[3] Gu Z, Tang X. Ordered regular equivalence relations on ordered hypersemigroups. Journal of Algebra 2016; 450: 384–397. doi:10.1016/j.jalgebra.2015.11.026
[4] Heidari D, Davvaz B. On ordered hyperstructures. “Politehnica” University of Bucharest. Scientific Bulletin. Series A. Applied Mathematics and Physics 2011; 73 (2): 85–96.
[5] Kehayopulu N. Remark on ordered semigroups. Mathematica Japonica 1990; 35 (2): 1061–1063.
[6] Kehayopulu N. On semilattices of simple poe-semigroups. Mathematica Japonica 1993; 38 (2): 305–318.
[7] Kehayopulu N. On ordered hypersemigroups given by a table of multiplication and a figure. Turkish Journal of Mathematics 2018; 42 (4): 2045–2060. doi:10.3906/mat-1711-53
[8] Kehayopulu N. On semilattice congruences on hypersemigroups and on ordered hypersemigroups. European Journal of Pure and Applied Mathematics 2018; 11 (2): 476–492.
[9] Kehayopulu N, Tsingelis M. On subdirectly irreducible ordered semigroups. Semigroup Forum 1995; 50 (2): 161–177. doi:10.1007/BF02573514
[10] Kehayopulu N, Tsingelis M. Pseudoorder in ordered semigroups. Semigroup Forum 1995; 50 (3): 389–392. doi:10.1007/BF02573534
[11] Petrich M. Introduction to Semigroups. Charles E. Merrill Publ. Comp. A Bell & Howell Comp. Columbus, Ohio 1973; viii+198 pp.
[12] Tang J, Feng X, Davvaz B, Xie X-Y. A further study on ordered regular equivalence relations in ordered semihypergroups. Open Mathematics 2018; 16 (1): 168–184. doi:10.1515/math-2018-0016
[13] Tang J, Luo Y, Xie X-Y. A study on (strong) order-congruences in ordered semihypergroups. Turkish Journal of Mathematics 2018; 42 (3): 1255–1271. doi:10.3906/mat-1512-83
[14] Xie, X-Y. On regular, strongly regular congruences in ordered semigroups. Semigroup Forum 2000; 61 (2): 159–178. doi:10.1007/PL00006018