ϕ-(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 rkn−k+1 1 ∈ r, g(rkn−k+1 1 ) ∈ q − {0} implies that there are (k − 1)n − k + 2 of the r, is whose g-product is in q ( g(r(k−1)n−k+2 1 ) ∈ q or a g-product of (k − 1)n − k + 2 of r, is ,except g(r(k−1)n−k+2 1 ), is in r(m,n)(q) ) . in this paper, we aim to extend the notions to the concepts of ϕ-(k, n)-absorbing and ϕ-(k, n)-absorbing primary hyperideals. assume that ϕ is a function from hi(r) to hi(r) ∪ {φ} such that hi(r) is the set of hyperideals of r and k is a positive integer. we call a proper hyperideal q of r a ϕ-(k, n)-absorbing (primary) hyperideal if for rkn−k+1 1 ∈ r, g(rkn−k+1 1 ) ∈ q − ϕ(q) implies that there are (k − 1)n − k + 2 of the r, is whose g-product is in q ( g(r(k−1)n−k+2 1 ) ∈ q or a g-product of (k − 1)n − k + 2 of r, is ,except g(r(k−1)n−k+2 1 ), is in r(m,n)(q) ) . several properties and characterizations of them are presented.