\phi-(k,n)-absorbing (primary) hyperideals in a Krasner (m,n)-hyperring

Various expansions of prime hyperideals have been studied in a Krasner (m,n)-hyperring R. For instance, a proper hyperideal Q of R is called weakly (k,n)-absorbing (primary) provided that for r_1^{kn-k+1} \in R, g(r_1^{kn-k+1}) \in Q-\{0\} implies that there are (k-1)n-k+2 of the r_i^,s whose g-product is in Q \Bigl ( g(r_1^{(k-1)n-k+2}) \in Q or a g-product of (k-1)n-k+2 of r_i^,s ,except g(r_1^{(k-1)n-k+2}), is in \boldsymbol{ r}^{(m,n)}(Q) \Bigr ). In this paper, we aim to extend the notions to the concepts of \phi-(k,n)-absorbing and \phi-(k,n)-absorbing primary hyperideals. Assume that \phi is a function from \mathcal{HI}(R) to \mathcal{HI}(R) \cup \{\varnothing\} such that \mathcal{HI}(R) is the set of hyperideals of R and k is a positive integer. We call a proper hyperideal Q of R a \phi-(k,n)-absorbing (primary) hyperideal if for r_1^{kn-k+1} \in R, g(r_1^{kn-k+1}) \in Q-\phi(Q) implies that there are (k-1)n-k+2 of the r_i^,s whose g-product is in Q \Bigl ( g(r_1^{(k-1)n-k+2}) \in Q or a g-product of (k-1)n-k+2 of r_i^,s ,except g(r_1^{(k-1)n-k+2}), is in \boldsymbol{ r}^{(m,n)}(Q) \Bigr ). Several properties and characterizations of them are presented.

\phi-(k,n)-absorbing (primary) hyperideals in a Krasner (m,n)-hyperring Keywords:

\phi-(k , n)-absorbing hyperideal , \phi-(k , n)-absorbing primary hyperideal