Regular elements and Green's relations in Menger algebras of terms

Defining an (n+1)-ary superposition operation ${\displaystyle S^n}$ on the set ${\displaystyle W_{\tau }\left(X_n\right)}$ of all n-ary terms of type τ, one obtains an algebra ${\displaystyle n-clo\ne \tau :=(W_{\tau }\left(X_n\right);S^n,x_1,...,x_n)}$ of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation ${\displaystyle S^n}$ there are different possibilities to define binary associative operations on the set ${\displaystyle W_{\tau }\left(X_n\right)}$ and on the cartesian power ${\displaystyle W_{\tau }{\left(X_n\right)}^n}$. In this paper we study idempotent and regular elements as well as Green's relations in semigroups of terms with these binary associative operations as fundamental operations.