Some definition :
Let $S$ be an inverse semigroup and $E$ be the set of all idempotents in $S$. Then $E$ form a semilattice under this relation $$e \leq f \; \; \text {iff} \; ef = fe = e$$ Define a relation $ \huge{\mu}$ $= \{ (e,f) \in E \times E : Ee \cong Ef \}$, where $Ee$ is the principal ideal generated by $e$ in $E$ and define a set $T_{e,f}$ consist of all isomorphism from $Ee$ onto $Ef$ if $(e,f) \in \huge \mu$.
Define $T_E = \underset{(e,f) \in \huge \mu}{\huge\cup} T_{e,f}$ which form a inverse semigroup under the composition.
A semigroup $S$ is said to be bisimple if $S$ has only one $\mathcal D$- class, where $\mathcal D$ is a Green's relation.
A subsemigroup $T$ of $T_E$ is said to be transitive if $\forall e,f \in E,$ $$ T \cap T_{e,f} \neq \phi$$
My question is
Suppose $T_E$ is bisimple semigroup. Is it true that ever subsemigroup $T$ of $T_E$ is transitve.
Any help would be appreciated. Thank you.