The purpose of the article is to study about spherical cubic sets and spherical cubic bi-ideals of Gamma near-ring R. We define spherical internal and external cubic sets and their properties. We discuss P-order and R-order, P-union, P-intersection, R-union and R-intersection of spherical cubic sets. We define spherical cubic bi-ideals of gamma near-ring R and prove that P-union, P-intersection, R-union and R-intersection of spherical cubic bi-ideals of Gamma near-ring R are also spherical cubic bi-ideals of Gamma near-ring R.
Motivated by the theory of 0-simple semigroup and M -semigroup, in this paper we introduce the notion of 0-simple M -semigroup. We also focus our research towards soft set theory and introduce the notion of soft Msemigroup and provide some results on it. In a 0-simple M -semigroup, we prove that 0-minimal ideal is a 0-simple M -semigroup and in soft setting, we prove that the union of the two soft M -semigroups is also a soft M -semigroup with the condition that the intersection of the subsets are empty. Also we justify that the Cartesian product of the two soft M -semigroups is also a soft M -semigroup.
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