Example of a semigroup satisfy some properties

Question

I want to find an example a finite semigroup $S$ and $K \subseteq S$ satisfy the properties

• For any $a,b \in K$, we have $a,b \in \langle c \rangle$ for some $c \in S$.

• $K$ does not hold closure property.

Thanks for any kind of help

Answer 1

Minimal example: take the semigroup $S = \{1, 2, 3\}$ equipped with the operation $x* y = \min\{x+y, 4\}$ and $K = \{2\}$. Then $2 = 1 * 1$, but $2 * 2 = 3$.

Answer 2

Let $S=<\mathbb Z_6,+>$ and $K=<\{1,5\},+>$.