%0 Journal Article
%T $(m,n)$-Hyperideals in Ordered Semihypergroups
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Mahboob, Ahsan
%A Khan, Noor Mohammad
%A Davvaz, Bijan
%D 2020
%\ 01/01/2020
%V 12
%N 1
%P 43-67
%! $(m,n)$-Hyperideals in Ordered Semihypergroups
%K Ordered semihypergroups
%K $(m,0)$-hyperideals
%K $(0,n)$-hyperideals
%R 10.29252/cgasa.12.1.43
%X In this paper, first we introduce the notions of an $(m,n)$-hyperideal and a generalized $(m,n)$-hyperideal in an ordered semihypergroup, and then, some properties of these hyperideals are studied. Thereafter, we characterize $(m,n)$-regularity, $(m,0)$-regularity, and $(0,n)$-regularity of an ordered semihypergroup in terms of its $(m,n)$-hyperideals, $(m,0)$-hyperideals and $(0,n)$-hyperideals, respectively. The relations ${_m\mathcal{I}}, \mathcal{I}_n, \mathcal{H}_m^n$, and $\mathcal{B}_m^n$ on an ordered semihypergroup are, then, introduced. We prove that $\mathcal{B}_m^n \subseteq \mathcal{H}_m^n$ on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. We also show that the $(m,0)$-regularity [$(0,n)$-regularity] of an element induce the $(m,0)$-regularity [$(0,n)$-regularity] of the whole $\mathcal{H}_m^n$-class containing that element as well as the fact that $(m,n)$-regularity and $(m,n)$-right weakly regularity of an element induce the $(m,n)$-regularity and $(m,n)$-right weakly regularity of the whole $\mathcal{B}_m^n$-class and $\mathcal{H}_m^n$-class containing that element, respectively.
%U https://cgasa.sbu.ac.ir/article_87415_1fd525cccd124d58a33309087242f95f.pdf