Graph Representations of a Free Clifford Monoid and a Free Semilattice with Identity

The following conclusion is shown that a free Clifford monoid $C_{X}$ on X is isomorphic to a monoid $\overline{B_{X}}$ obtained by a subset of $B_{X}$, which is the set of all birooted word trees on the same X, together with a new multiplication. Furthermore, the relation between $B_{X}$ and $\overline{B_{X}}$ is considered. Also, it is shown that there is an isomorphism from a free semilattice with identity $Y_{X}$ on X onto a monoid $\overline{T_{X}}$ obtained by a subset of $T_{X}$, which is the set of rooted word trees on the same X.