I would you like to construct an isomorphism with generalized full transformations.
All case for finite sets.
- $\textbf{Case I}:$ Let $\theta:Y\rightarrow X_n$ is a bijection. Then
$S=\mathfrak{T}(X_n, Y,\theta)$ is a generalized full transformation semigroup with sandwich operation $\alpha\star\beta=\alpha\circ\theta\circ\beta$
$T=\mathfrak{T}(X_n, \circ)$
Then $$\phi:S\rightarrow T, \alpha\mapsto \alpha\theta$$ is an isomorphism.
By thinking about how can we construct $\phi$ I try to find $T$.
- $\textbf{Case II}:$ Let $\theta:Y\rightarrow X_n$ is a $1-1$ and $\mid Y\mid=m<n$ (i.e., Let $\theta$ not be onto.)
Then $S=\mathfrak{T}(X_n, Y,\theta)$ and by taking $$\phi:S\rightarrow T, \alpha\mapsto \alpha\theta$$ Do it $\phi$ be an isomorphism we find $T=\{\alpha\in T_n: im(\alpha)\subseteq im(\theta)\}$
My question related to Case III:
- $\textbf{Case III}:$ Let $\theta:Y\rightarrow X_n$ is onto and $\mid Y\mid=m>n$.
Let $S=\mathfrak{T}(X_n, Y,\theta)$,
How can we construct $\phi$ for which $T$?
Thank you for help!