Ternary Menger Algebras: A Generalization of Ternary Semigroups

Abstract

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.

v-Regular Ternary Menger Algebras and Left Translations of Ternary Menger Algebras

“…We call a ternary Menger algebra of rank n with all elements are left zero elements as a left zero ternary Menger algebra of rank n. (i) Consider on the set of all positive real numbers ( [45]) R + together with a (2n + 1)-ary operation • defined by…”

“…where • is the usual (binary) multiplication. Then, (R + , •) forms a ternary Menger algebra of rank n. (ii) Let R be the set of all real numbers ( [45]). Define a (2n + 1)-ary operation…”

“…. , y n ) ( [45]). Then the based set G together with a (2n + 1)ary operation •, which is defined by •(x, y 1 , .…”

“…Up to 2010, Santiago, M. L. and Sri Bala, S. [27] investigated some interesting properties of regular ternary semigroups. Furthermore, there were interesting results related to ternary semigroups and regular ternary semigroups (see [28][29][30][31]).…”

“…According to the algebraic structure of Menger algebras of rank n, the authors discovered that such an algebraic structure is a generalization of (binary) semigroups, while it is not a generalization of arbitrary ternary semigroups. Based on this important result, the notion of ternary Menger algebras of rank n where n is a fixed natural number was first introduced by the authors in 2021 (see [45]). In particular, the isomorphism theorems and the reduction of ternary Menger algebras of rank n into Menger algebras of rank n were investigated.…”