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An investigation on S-hypersystems over semihypergroups

In this paper, we first explain the essential definition of S-hypersystem A, which means there exists a good homomorphism from a semihypergroup S to a semihypergroup M (composed of all mappings from A to P(A)), and give the fundamental normal homomorphism theorems on S-hypersystems. In addition, we introduce the concept of the category of S-hypersystems and study the product and coproduct of S-hypersystems. Finally, we prove that every S-pohypersystem is a subdirect product of subdirectly irreducible S-pohypersystems.