Let n be a fixed natural number. Menger algebras of rank n, 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 n, 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 n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented.

Let n be a fixed natural number. Ternary Menger algebras of rank n, which was established by the authors, can be regarded as a suitable generalization of ternary semigroups. In this article, we introduce the notion of v-regular ternary Menger algebras of rank n, which can be considered as a generalization of regular ternary semigroups. Moreover, we investigate some of its interesting properties. Based on the concept of n-place functions (n-ary operations), these lead us to construct ternary Menger algebras of rank n of all full n-place functions. Finally, we study a special class of full n-place functions, the so-called left translations. In particular, we investigate a relationship between the concept of full n-place functions and left translations

The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. Then, we extend the well-known result on group theory and semigroup theory, the so-called Cayley’s theorem, to study on ternary semihypergroups. This leads us to construct the ternary semihypergroups of all multivalued full binary functions. In particular, we investigate that each element of a ternary semihypergroup induced by a semihypergroup can be represented by a multivalued full binary function. Moreover, we introduce the concept of translations for ternary semihypergroups which can be considered as a generalization of translations on ternary semigrgoups. Then, we construct ternary semihypergroups of all multivalued full functions and ternary semihypergroups via translations. So, some interesting algebraic properties are investigated. At the last section, we discover that there are ternary semihypergroups satisfying some significant conditions which can be reduced to semihypergroups. Furthermore, ternary semihypergroups with another one condition can be reduced to idempotent semihypergroups.