| Summary: | Let <i>n</i> be a fixed natural number. Menger algebras of rank <i>n</i>, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank <i>n</i>, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank <i>n</i> and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank <i>n</i>. These lead us to study the quotient ternary Menger algebras of rank <i>n</i> and to investigate the homomorphism theorem for ternary Menger algebra of rank <i>n</i> with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented.
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