Γ-semihyperrings: Approximations and rough ideals

The notion of rough sets was introduced by Z. Pawlak in 1982. The concept of Γ-semihyperring is a generalization of semihyperring, a generalization of Γ-semiring and a generalization of semiring. In this paper, we study the notion of a rough (rough prime) ideal in a Γ-semihyperring. Also, we discuss the relation between the upper and lower rough ideals and the upper and lower approximation of their homomorphism images. 1 Preliminaries and basic definition The process of analyzing data under uncertainty is a main goal for many real life problems. The present century is distinguished by the tendency of using the available data in the process of decision making. The real data derived from actual experiments needs a special treatment to get information more close to reality. Pawlak [16], introduced the rough set theory, which is an excellent tool to handle a granularity of data. In rough set theory, given an equivalence relation on a universe, we can define a pair of rough approximations which provide a lower bound and an upper bound for each subset of the universe set. Biswas and Nanda [3] defined the notion of rough subgroup. Kuroki [14], introduced the notion of a rough ideal in a semigroup, studied approximations of a subset in a semigroup and discussed the product structures of rough ideals. In [15], Kuroki and Wang provided some 2000 Mathematics Subject Classification: 20N20, 16Y99