‎$‎J‎$‎-hyperideals and their expansions in a Krasner ‎$‎(m,n)‎$‎-hyperring

Over the years, different types of hyperideals have been introduced in order to let us fully realize the structures of hyperrings in general. ‎ ‎The aim of this research work is to define and characterize a new class of hyperideals in a Krasner ‎$‎(m,n)‎$‎-hyperring that we call ‎n-ary ‎‎$‎J‎$‎-hyperideals. A proper hyperideal ‎$‎Q‎$ o‎f a ‎Krasner ‎$‎(m,n)‎$‎-hyperring with the scalar identity ‎$‎1_R‎$ ‎is said to be an n-ary ‎$J‎$‎-hyperideal ‎‎‎‎if whenever $x_1^n \in R$ such that ‎$‎g(x_1^n) ‎\in ‎Q‎$ ‎and ‎‎$‎x_i \notin J_{(m,n)}(R)$, then ‎$‎g(x_1^{i-1},1_R,x_{i+1}^n) \in Q‎$‎. Also, we study the concept of n-ary ‎$‎\delta‎$‎-‎$‎J‎$‎-hyperideals as an expansion of n-ary $‎J‎$‎-hyperideals. Finally, we extend the notion of n-ary $‎\delta‎$‎-‎$‎J‎$‎-hyperideals to ‎$‎(k,n)‎$‎-absorbing ‎$‎\delta‎$‎-‎$‎J‎$-hyperideals.‎ ‎Let $\delta$ be a hyperideal expansion of a Krasner $(m,n)$-hyperring $R$ and $k$ be a positive integer‎. ‎A proper hyperideal $Q$ of $R$ is called $(k,n)$-absorbing $\delta‎$‎-‎$‎J‎$‎-hyperideal if for $x_1^{kn-k+1} \in R$‎, ‎$g(x_1^{kn-k+1}) \in Q$ implies that $g(x_1^{(k-1)n-k+2}) \in J_{(m,n)}(R)$ or a $g$-product of $(k-1)n-k+2$ of $x_i^,$ s except $g(x_1^{(k-1)n-k+2})$ is in $\delta(Q)$‎.