Example of inverse semigroup with at least two idempotent elements We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define  $$s\sim t\Leftrightarrow \exists\ e\in E(S):\ se=te,\qquad (s,t\in S); $$
and suppose that $[s]=\{t\in S:\ s\sim t\}$.
I have two request:


*

*Could we say that $[s]=\{se: e\in E(S)\}$ for any $s\in S$?

*I need an inverse semigroup $S$ such that $S\neq E(S)$ and $E(S)$ has at least two elements.


Actually, if you take any inverse semigroup $S$ with a zero element $0$, all elements of $S$ will be equivalent for $\sim$ since $s0 = 0 = t0$ for all $s, t \in S$. Thus $[0] = S$ but $\{0e \mid e \in E(S)\} = \{0\}$.
Let $S = \{a, b, ab, ba, 0\}$ be the semigroup defined by the relations $aba = a$ and $bab = b$ and $0x = 0 = x0$ for all $x \in M$. Then $S$ is an inverse semigroup with zero, $E(S) = \{ab, ba, 0\} \not= S$ and $|E(S)| \geqslant 2$.
