Groupoids and the associative law. VII. Semigroup distance of SH-groupoids
By a relational system we mean a couple where is a set and is a binary relation on , i.e. . To every directed relational system we assign a groupoid on the same base set where if and only if . We characterize basic properties of by means of identities satisfied by and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.
A family of loops is studied, which arises with its binary operation in a natural way from some transversals possessing a ``normality condition''.
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.