A Study of Congruence on (&amp;lt;i&amp;gt;n, m&amp;lt;/i&amp;gt;)-semigroup

Jiangping Xiao

doi:10.11648/j.pamj.20170604.13

Abstract

Congruence is a special type of equivalence relation which plays a vital role in the study of quotient structures of different algebraic structures. The purpose of this paper is to study the quotient structure of (n, m)-semigroup by using the notion of congruence in (n, m)-semigroup. Firstly, the concept of homomorphism on (n, m)-semigroup is introduced. Then, the concept of congruence on (n, m)-semigroup is defined, and some basic properties are studied. Finally, it is proved that the set of congruences on an (n, m)-semigroup is a complete lattice. All these generalize the corresponding notions and results for usual binary and ternary semigroups.

Highlights

An, -semigroup, is a nonempty set with an associative mapping : →
As is know to all, congruence is a special type of equivalence relation which plays a vital role in the study of quotient structures of different algebraic structures
It is proved that the set of congruences on an, -semigroup is a complete lattice

Summary

Introduction

An , -semigroup , is a nonempty set with an associative mapping :. 2, 1 -semigroup is a usual binary semigroup. There are several papers study the , -semigroup that can be found in [2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 17, 19, 20, 21, 22, 23]. Green’s relations and congruences are two important tools for studying semigroup algebraic structure. -semigroups as a generalization of binary and ternary semigroups had been extensively studied. As is know to all, congruence is a special type of equivalence relation which plays a vital role in the study of quotient structures of different algebraic structures. The aim of this paper is to generalize some definitions and results of congruences on usual binary and ternary semigroups to , -semigroup. It is proved that the set of congruences on an , -semigroup is a complete lattice

Preliminaries

Findings

Conclusion