On Primary and Regular $\Gamma$-Semihypergroups

Abstract

In this paper the notions of primary and semiprimary $\Gamma$-hyperideals in $\Gamma$-semihypergroup are introduced. It is shown that if $rad.(I)$ is a maximal $\Gamma$-hyperideal of $S$ then $I$ is a semiprimary $\Gamma$-hyperideal of $S$. It is proved that a $\Gamma$-semihypergroup $S$ is semiprimary if and only if prime $\Gamma$-hyperideals of $S$ forms a chain under set inclusion. Regular set in $\Gamma$-semihypergroup and regular $\Gamma$-hypergroup are defined with several examples. Characterizations of regular $\Gamma$-semihypergroup is established in terms of left and right $\Gamma$-hyperideals.