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Turkish Journal of Mathematics

DOI

10.3906/mat-2108-78

Abstract

If $(S,\circ,\le)$ is an ordered hypersemigroup, an equivalence relation $\rho$ on $S$ is called congruence if $(a,b)\in\rho$ implies $(a\circ x, b\circ x)\in\rho$ and $(x\circ a, x\circ b)\in\rho$ for every $x\in S$; in the sense that for every $u\in a\circ x$ there exists $v\in b\circ x$ such that $(u,v)\in\rho$ and for every $u\in x\circ a$ there exists $v\in x\circ b$ such that $(u,v)\in\rho$. It has been proved in Turk J Math 2021(5) [On the paper "A study on (strong) order-congruences in ordered semihypergroups"] that if $S$ is an ordered hypersemigroup, then there exists a congruence $\rho$ on $S$ such that $S/\rho$ is an ordered hypersemigroup. This result, is the main result for an involution ordered hypersemigroup by Xinyang Feng, Jian Tang and Yanfeng Luo in U.P.B. Sci. Bull, Series A, 2018, but its proof is wrong; the correct proof is given in the present paper. Examples illustrate the results.

Keywords

ordered hypersemigroup, involution, quasi pseudoorder, congruence, pseudoorder, strong congruence

First Page

2466

Last Page

2476

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