Example of inverse semigroup with at least two idempotent elements

Question

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:

1. Could we say that $[s]=\{se: e\in E(S)\}$ for any $s\in S$?
2. I need an inverse semigroup $S$ such that $S\neq E(S)$ and $E(S)$ has at least two elements.

Answer 1

We know that the set of natural number with multiplication defined by $m\cdot n=\min\{m,n\}$ is an inverse semigroup. Now suppose that $G$ is a group such that there is an element $h\in G$ such that $h^2\neq h$ and suppose that $e$ is its identity. Obviously $S:=N\times G$ is an inverse semigroup with mutiplication $$(n,g)(m,k)=(\min\{n,m\}, gk).$$ Now because $(1,h)(1,h)=(1,h^2)\neq (1,h)$ we get $S\neq E(S)$. Also for any $n\in N$ we have $$(n,e)(n,e)=(\min\{n,n\},e^2)=(n,e)\in E(S)$$ which shows that $|E(S)|\geq 2$.

From the previous part $E(S)=\{(n,0): n\in N\}$ and $[(m,q)]=\{(n,q):n\in N\}$. Also $$\{(2,g)E: E\in E(S)\}=\{(2,g)(n,0): n\in N\}=\{(1,g),(2,g)\}$$ but $(2,g)\sim (3,g)$.

Answer 2

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 $|E(S)| \geqslant 2$.