Structure of the (Total) Transformation Monoids Under Rank N Generators

Throughout this paper our study focuses on transformation semigroups. These kinds of semigroups are the corner stone of semigroup theory. This is because every semigroup is isomorphic to transformation semigroup. The (total) transformation monoid $T_{X_n}$ on a finite set $X_n=\{1,2,\dots ,n\}$ where $n\ge 0$ , $n\in Z$ , is a semigroup of mapping that takes a set $X_n$ into itself, under the operation of composition of mapping with identity $I_{X_n}$ . In this paper, we use an algebraic method for considering the monoid $T_{{\left(\mathrm{Fl}\right)}_n\left(G\right)}$ , where an independence algebra ${\left(\mathrm{Fl}\right)}_n\left(G\right)$ is a disjointed union of sets of the form $G{x}_i$ for all 1 $\le i\le n.$ Firstly, particular attention is paid to find the isomorphism between $T_{{\left(\mathrm{Fl}\right)}_n\left(G\right)}$ and the endomorphism monoid $\mathrm{End}{\left(F\ell \right)}_n\left(G\right).$ Secondly, the embeddedness of $T_{{\left(\mathrm{Fl}\right)}_n\left(G\right)}$ in (full) wreath product of $T_n$ by $G^n$ has been found. Finally, the description of Green's relation of $T_{{\left(\mathrm{Fl}\right)}_n\left(G\right)}$ has been provided.