A Note on Relative Ordered (m,n)-Hyperideals in Ordered Semihypergroups
Author : Dr. Abul Basar, Dr. Poonam Kumar Sharma, Dr. Ayaz Ahmad and Dr. Bhavanari Satyanarayana
Abstract :
In this paper, we study relative ordered (m, n)-hyperideals in ordered semihypergroups. We also study relative (m, 0)-hyperideals and relative (0, n)-hyperideals as well as characterize regular ordered semihypergroups, and obtain some results based on these relative hyperideals. We prove that the intersection of all relative ordered (m, n)-hyperideals of S containing s is a relative ordered (m, n)-hyperideal of S containing s. Suppose that (S,◦,≤) is an ordered semihypergroup, A ⊆ S and m,n are positive integers. We prove that if R(m,0) and L(0,n) be the set of all relative ordered (m,0)-hyperideals and the set of all relative ordered (0,n)-hyperideals of S, respectively. Then the following assertions are true: i) S is relative (m,0)-regular if and only if for all R ∈ R(m,0), R = (Rm ◦A]A. ii) S is relative (0,n)-regular if and only if for all L ∈ R(0,n), L = (A◦Ln]A. Furthermore, suppose that (S, ◦, ≤) is an ordered semihypergroup and m, n are non-negative integers. Let A ⊆ S. Suppose that A(m,n) is the set of all relative ordered (m, n)-hyperideals of S. Then, we have the following: S is (m,n)-regular ⇐⇒ ∀A ∈ A(m,n),A = (Am ◦A◦An]A.
Keywords :
Ordered semihypergroup, regular ordered semihypergroup, relative ordered bi-hyperideal, relative ordered (m, n)-hyperideal
