ABSTRACT.

In this paper, we introduce the concept of ordered quasi-hyperideals of regular ordered semihypergroups, and study the basic results on ordered quasi-hyperideals of ordered semihypergroups. We also investigate regular ordered semihypergroups in terms of its ordered quasi-hyperideals, ordered right hyperideals and ordered left hyperideals. We prove that:

(i) A partially ordered semihypergroup \( S \) is regular if and only if for every ordered bi-hyperideal \( B \), every ordered hyperideal \( I \) and every ordered quasi-hyperideal \( Q \), we have
\[
B \cap I \cap Q \subseteq (B \circ I \circ Q),
\]

and

(ii) A partially ordered semihypergroup \( S \) is regular if and only if for every ordered quasi-hyperideal \( Q \), every ordered left hyperideal \( L \) and every ordered right-hyperideal \( R \), we have
\[
R \cap Q \cap L \subseteq (R \circ Q \circ L).
\]