Abstract
In this paper, we consider a theory of elements u of a groupoid ( X , * ) that are associated with certain functions u ^ : X → X , pseudo-inverse functions, which are generalizations of the inverses associated with units of groupoids with identity elements. If classifying the elements u as special of one of twelve types, then it is possible to do a rather detailed analysis of certain cases, leftoids, rightoids and linear groupoids included, which demonstrates that it is possible to develop a successful theory and that a good deal of information has already been obtained with much more possible in the future.
Highlights
Bruck [1] published a book, A survey of binary systems discussed in the theory of groupoids, loops and quasigroups, and several algebraic structures
This shows that every element u of a right-zero semigroup is LL-special and every function ub : X → X is a pseudo inverse function of u
That may exist for arbitrary binary systems
Summary
Bruck [1] published a book, A survey of binary systems discussed in the theory of groupoids, loops and quasigroups, and several algebraic structures. Some interesting results in groupoids were investigated by several researchers [3,4,5,6]. The notion of d-algebras, which is another useful generalization of BCK-algebras, was introduced by Neggers and Kim [8], and some relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs were investigated. Kim and Neggers [15] introduced the notion of Bin( X ), which is the collection of all groupoids defined on a set. It turns out that if we classify the elements u as special of one of twelve types, it is possible to do a rather detailed analysis of certain cases, leftoids, rightoids and linear groupoids included, which demonstrates that it is possible to develop a successful theory and that a good deal of information has already been obtained with much more possible in the future
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