Abstract
The concept of a semihypergroup is a generalization of the concept of a semigroup. As we know, in a semigroup, the composition of two elements is an element, while in a semihypergroup, the composition of two elements is a non-empty set. Indeed, semihypergroups are the simplest algebraic hyperstructures which possess the properties of closure and associativity. They are very important in the theory of sequential machines, formal language, and in certain applications. This chapter begins with history of algebraic hyperstructures and some basic results concerning semihypergroups. Then, we study regular semihypergroups, subsemihypergroups, hyperideals, quasi-hyperideals, prime and semiprime hyperideals, homomorphisms, and regular and strongly regular relations. In the rest of this chapter we present the notions of simple and cyclic semihypergroups.
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