Abstract

There are several equivalent axioms, which can be used to characterize the positive implicativity in B C K -algebras. In this paper, we investigate interrelationships among such axioms in a more general setting of groupoids, and several aspects regarding their differences in the theory of groupoids.

Highlights

  • Bruck ([1]) published a book, A survey of binary systems discussed in the theory of groupoids, loops and quasigroups, and several algebraic structures

  • Some notions of the graph theory were applied to the theory of groupoids ([7])

  • We proved that the collection of all positive implicative groupoids forms a variety, and discussed some relations between selective groupoids and the graph theory

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Summary

Introduction

Bruck ([1]) published a book, A survey of binary systems discussed in the theory of groupoids, loops and quasigroups, and several algebraic structures. There are several different axioms, which can give equivalent characterizations of the positive implicativity in BCK-algebras. By using other axioms or their induced results, the proofs of their equivalences were obtained in BCK-algebras. It is interesting and useful to investigate these axioms in a more general mathematical structure called groupoids. We discuss some relations among such axioms in groupoids, and obtain some results disclosing their differences in the groupoid setting. If we discuss these conditions in BCK/BCI-algebras, their delicate differences may not be discovered. Simple mathematical structures are difficult to deal with in some cases, they can capture essential ideas of some axioms, and provide a starting point of new mathematical structures in future

Preliminaries
Positive Implicativity
Central and Meet Implicativity
Quasi Implicativity
Medial and Positive Implicative Groupoids
Conclusions
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