Give an example of a language $L$ where $\min(\max L)\neq \max(\min L)$

Question

Give an example of a language $L$ where $\min(\max L)\neq \max(\min L)$.

I thought of the following language $L=\{a,bc, abc\}$.

$$ \min L=\{a,bc\}, \max L = \{abc\} $$ Then: $$ \min(\max L)=\min (\{abc\})=\{abc\}\neq \max(\min L)=\max(\{a,bc\})=\{a,bc\} $$ This seems too simple so I'm wondering if it's correct.

---

The definitions of $\max, \min$: $$ \min L= \{x|x\in L \land \text{there doesn't exist a non-empty substring }y \text{ of } x \text{ such that } y\in L \}\\ \max L = \{x|x\in L \land \lnot \exists y: xy\in L, y\neq \epsilon\} $$

Answer 1

Your example is perfect.

Being simple is an advantage and not a problem ;)

Answer 2

Now with the definitions it is clear that your example is right. However, your reasoning is not correct. $bc$ can be extended to the left by $a$ to form a longer string in the language, but there is no extension to the right. The definition of $max$ only asks for extensions to the right. Therefore $bc$ is also in the $max$, and your example works like this:

$$ \min(\max L)=\min (\{bc, abc\})=\{bc\}\ \ \neq\ \ \{a,bc\}=\max(\{a,bc\})=\max(\min L) $$