Example of a semigroup with unique idempotent which is not a monoid

Question

I am searching for an example of a semigroup, with unique idempotent element, that is not a monoid. Please help.

Answer 1

How about a 2-element null semigroup? If the elements are $x$ and $y$, then the multiplication can be $xy=y=yx=y^2=x^2$.

Answer 2

Ok, so let us go for something famous: as you know $(\mathbb{N},+)$ and $(\mathbb{N},\cdot)$ are both (commutative) monoids.

$(\mathbb{N},+)$ has one only idempotent, the identity: $0+0=0$,

$(\mathbb{N},\cdot)$ instead has two: $0\cdot 0=0$ and $1\cdot 1=1$. $1$ is the identity here and, if we take it away from $\mathbb{N}$, we observe that $\mathbb{N}\setminus\{1\}$ is again closed under multiplication.

Hence $(\mathbb{N}\setminus\{1\},\cdot)$ is a commutative semigroup which is not a monoid, with only one idempotent element, $0$, which is not the identity.

At the same way, the multiplicative structure over $\mathbb{N}\setminus\{1,\ldots,k-1,k\}$ is a semigroup with the above properties for all $k\ge 2$.