Fuzzy sets and non-complete 1-hypergroups.
We introduce the concept of a hyper BCI-algebra which is a generalization of a BCI-algebra, and investigate some related properties. Moreover we introduce a hyper BCI-ideal, weak hyper BCI-ideal, strong hyper BCI-ideal and reflexive hyper BCI-ideal in hyper BCI-algebras, and give some relations among these hyper BCI-ideals. Finally we discuss the relations between hyper BCI-algebras and hyper groups, and between hyper BCI-algebras and hyper -groups.
In this paper, we firstly introduce a special congruence relation U(μ, t) induced by a fuzzy ideal μ in a semigroup S. Then we define the lower and upper approximations based on a fuzzy ideal in semigroups. We can establish rough semigroups, rough ideals, rough prime ideals, rough fuzzy semigroups, rough fuzzy ideals and rough fuzzy prime ideals according to the definitions of rough sets and rough fuzzy sets. Furthermore, we shall consider the relationships among semigroups and rough semigroups,...