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Turkish Journal of Mathematics

DOI

10.3906/mat-2106-7

Abstract

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 $\rho$ an equivalence relation on S. Then $(i)$ If $\rho$ is a congruence, then $(S/\rho,\otimes)$ is a semihypergroup with respect to the hyperoperation $(a)_\rho\otimes (b)_\rho=\bigcup\limits_{c\in a*b} {(c)_\rho}$. $(ii)$ If $\rho$ is a strong congruence, then $(S/\rho,\otimes)$ is a semigroup with respect to the operation $(a)_\rho\otimes (b)_\rho=(c)_\rho$ for all $c\in a*b$.} The property (i) of the paper is certainly wrong as $\bigcup\limits_{c\in a*b} {(c)_\rho}$ is a subset of $S$ and not a nonempty subset of $S/\rho$ 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.

Keywords

Ordered hypersemigroup, congruence, pseudoorder, quasi pseudoorder, strong congruence

First Page

2035

Last Page

2049

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