Hyper BZ-algebras and semihypergroups

In this paper, we introduce the new concept of a hyper BZ-algebra which is a generalization of BZ-algebra and hyper BCI-algebra, and give some examples and basic properties. We discuss the relationships among hyper BZ-algebras, hyper BCC-algebras and hyper BCI-algebra. Moreover, we propose the concepts of antigrouped hyper BZ-algebras and generalized anti-grouped hyper BZ-algebras, and prove that the following important results: (1) Every anti-grouped hyper BZ-algebra is an antigrouped BZ-algebra; (2) Every generalized anti-grouped hyper BZ-algebra corresponds to a semihypergroup. Finally, we present a method to construct a new hyper BZ-algebra by using a hyper BCC-algebra and a standard generalized anti-grouped hyper BZ-algebra.