Halfnets and partial -loops
This paper deals with the origins and early history of loop theory, summarizing the period from the 1920s through the 1960s.
A groupoid is a homomorphic image of a subdirectly irreducible groupoid (over its monolith) if and only if has a smallest ideal.
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.