Quaternary Rectangular Bands and Representations of Ternary Semigroups

Abstract

The article is devoted to investigation of algebraic ternary structures. We start first by recalling the notion of ternary semigroups and its algebraic properties. Analogous to the concept of rectangular bands in ordinary semigroups, we define the new concept of quaternary rectangular bands, and investigate some of its algebraic properties. Based on the well-known results on ordinary semigroups, the so-called Cayley’s theorem and the full (unary) transformations are defined. These lead us to con-struct a new algebraic ternary structure and its ternary operation the so-called the ternary semigroup of all full binary transformations and the ternary composition via identity 1, respectively. Moreover, we prove that every abstract ternary semigroup induced by a semigroup can be represented by full binary transformations. Futhermore, the related algebraic property of the ternary semigroups with an identity element is investigated.