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

A hyper $BN$-algebra is a nonempty set $H$ together with a hyperoperation ``$\circledast$'' and a constant $0$ such that for all $x, y, z \in H$: $x \ll x$, $x \circledast 0 = \{x\}$, and $(x \circledast y) \circledast z = (0 \circledast z) \circledast (y \circledast x)$, where $x \ll y$ if and only if $0 \in x \circledast y$. 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.