A Study of Congruence on ( n, m )-semigroup

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.