Abstract
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.
Highlights
An algebraic structure ( G, o ) is called a Menger algebra of rank n if the (n + 1)-ary operation o, which is defined on G, satisfies the superassociative law, i.e., o (o ( x, y1, . . . , yn ), z1, . . . , zn ) = o ( x, o (y1, z1, . . . , zn ), . . . , o), Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
We present the notion of v-regular ternary Menger algebras of rank n, which can be considered as a generalization of regular ternary semigroups
By the definition of the set of all left translations Λ( T ) defined on a ternary Menger algebra ( T, ) of rank n, we obtain that Λ( T ) ⊆ T ( T n, T )
Summary
A generalization of regular semigroups, which is called v-regular Menger algebras of rank n, and its interesting properties were established and studied by Trokhimenko V. Similar to the concept of partial transformation on semigroups, Menger algebras F ( X n , X ) of all partial n-place functions were constructed Such an algebraic structure is the set F ( X n , X ) of all partial n-place functions together with the Menger’s superposition. We complete this section by showing a relationship between the set of all full n-place functions, the set of all left translations and left zero ternary Menger algebras of rank n. The conclusions and future works are provided in the last section
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