Annals of Communications in Mathematics

AbstractIn this paper, first we obtain the necessary and sufficient condition for an ordered Γ-semihypergroup H to be a relative completely regular 2-duo ordered Γ-semihypergroup for any relative (2, 2)-Γ-hyperideal as well as for any relative (2, 2)-quasi-Γ-hyperideal of H. Then, we find the necessary and sufficient condition that Q = (Q2]S for every relative (2, 2)-Γ-hyperideal Q of H to be a relative quasi-Γ-prime with S ⊆ H. Finally, we prove the necessary and sufficient condition for relative (2, 2)-Γ-hyperideal to be a relative quasi-Γ-prime for a relative completely regular and relative (2, 2)-Γ-hyperideal of H making a chain with inclusive relation.

AbstractIn this paper, the main goal is to study an ordered Γ-semihypergroup H in the context of the characterizations of the associated Γ-semihypergroup B(H) of all bi-Γ-hyperideals of H. We show that an ordered Γ-semihypergroup H is a Clifford ordered Γ-semihypergroup if and only if B(H) is a semilattice. We also show that a Γsemihypergroup B(H) is a normal band if and only if the ordered Γ-semihypergroup H is simultaneously regular and intra regular. Furthermore, for each subclass S with many bands, we prove that for an ordered Γ-semihypergroup H, the conditional inclusion B(H) ∈ S holds true.