Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101$(m,n)$-Hyperideals in Ordered Semihypergroups436787415ENAhsanMahboobAligarh Muslim UniversityNoor MohammadKhanAligarh Muslim UniversityBijanDavvazYazd UniversityJournal Article20190405In 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 ${_mmathcal{I}}, mathcal{I}_n, mathcal{H}_m^n$, 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.http://cgasa.sbu.ac.ir/article_87415_1fd525cccd124d58a33309087242f95f.pdf