Is $L=\left \{ wxw \,\,|\,\,w,x \,\in \left ( a+b \right )^{+}\right \}$ regular?
We can say that Language
$L_{2}=\left \{ wxw \,\,|\,\,w,x \,\in \left ( a+b \right )^{*}\right \}$ is regular
By making $x=(a+b)^{*}$ and $w=\epsilon$
Based on above Understanding , i can say that $L$ is regular?
My regular expression will be-: $L=a(a+b)^{+}a+b(a+b)^{+}b+a(a+b)^{+}b+b(a+b)^{+}a$ Am i right?
When is a string of the form $a^*ba^*b$ in the language $L$, i.e., of the form $wxw$, for nonempty $w,x$?
Or, to be more explicit, what is the language $L \cap a^*ba^*b$?
The answer previously provided here is wrong. Counter example is $ab a ab$, see comment section below
Your conclusions are correct, but your regular expression also accepts the strings $aab$ or $abb$, which are not in $L$.
$a(a + b)^+a + b(a + b)^+b$ accepts a string $s$ if and only if $s \in L$, so $L$ is regular.
