Looking at Two Ways of Constructing Quotient Hyper BN-algebras and Some Notes on Hyper BN-ideals

Abstract

A hyper $BN$-algebra is a nonempty set $H$ together with a hyperoperation ``$⊛$'' and a constant $0$ such that for all $x,y,z\in H$: $x\ll x$, $x⊛0=\{x\}$, and $(x⊛y)⊛z=(0⊛z)⊛(y⊛x)$, where $x\ll y$ if and only if $0\in x⊛y$. We investigated the structures of ideals in the Hyper $BN$-algebra setting. We established equivalency of weak hyper $BN$-ideals and hyper sub$BN$-algebras. Also, we found a condition when a strong hyper $BN$-ideal become a hyper $BN$-ideal. Finally, we looked at two ways in constructing the quotient hyper $BN$-algebras and investigated the relationship between the two constructions.