Ternary Menger Algebras: A Generalization of Ternary Semingroups

Abstract

The notion of ternary semigroups was discovered S. Banach. In algebraic struc-ture of ternary semigroups, we can conclude that ternary semigroups are more extensive structure than semigroups which follows from the important remark: every semigroup can be induced to a ternary semigroup, while a ternary semigroup does not necessary reduce to a semigroup. Based on this knowledge, the interesting algebraic properties of ternary semigroups are stutied by many mathematicians. This thesis is devoted to investigation of ternary algebraic structure, and to con- struction of a new (2n + 1)-ary algebraic structure. Analogous to the concept of rectangu-lar bands on semigroups, we extend this concept to construct the quaternary rectangular bands, and the characterization of the quaternary rectangular bands. By using the con-cepts of full transformations and Cayley's theorem on semigroups, we define a new ternary algebraic structure and its ternary operation the so-called the ternary semigroups of all full binary transformations and the ternary composition via identity 1, respectively. Fur-thermore, Cayley's theorem for ternary semigroups is proved. The notion of Menger algebras of rank n, which can be considered as a general-ization of semigroups, was introduced by K. Menger in 1946. Based on this knowledge, the important question arises: what structure is a generalization of ternary semigroups? In this thesis, we introduce the notion of ternary Menger algebras of rank n, which is a generalization of ternary semigroups. Moreover, we establish the so-called a diagonal ternary semigroup and then investigate its interesting properties. Furthermore, we intro-duce the concept of homomorphsims and congruences on ternary Menger algebras. These lead us to study the quotient structure of ternary Menger algebras, and to investigate the isomorphsim theorems. Finally, the characterization of reduction of ternary Menger algebras into Menger algebras is presented.