Let $C_{\omega} = \{ e_0, e_1, e_2, \cdots \}$ with $$e_0 > e_1 > e_2 > \cdots > e_n > e_{n+1} > \cdots$$ form a semilattice.
We know that $C_{\omega}$ form a commutative semigroup under the binary operation $$ e_i \cdot e_j = e_i \; \; \text{iff} \; i \leq j. $$
A subset $I \subset S$, where $S$ is a semigroup is said to be an ideal of $S$ if $SI \subseteq I$ and $IS \subseteq I$.
The principal ideal generated by $e_k$ is of the form $C_{\omega}e_k = \{ e_k, e_{k+1}, \cdots \}$
I know that for each $m, n \in \mathbb N$, there is an isomorphism between $C_{\omega}e_n$ and $C_{\omega}e_m$ which is $$ \tau_{n,m} : C_{\omega}e_n \rightarrow C_{\omega}e_m$$ defined by $e_k \tau_{n,m} = e_{k-n+m}$ for all $k \geq n$.
My Question is :
How to show that there is no isomorphism between $C_{\omega}e_n$ and $C_{\omega}e_m$ other than $\tau_{n,m}$.