Given $m,n∈Z$, A is a finite alphabet set ,and $L=\{(a^m,a^n)\}^*$ is subset of $A^*\times A^*$ . Is this language regular ? For example, is $L=\{(a^3,a^7)\}^*$ regular ?
Here L is not the set $\{a^m,a^n\}^*$. $L$ is not a unitary set ,but a binary set, which is a language over $\{A∪\{\$\}\}\times \{A∪\{\$\}\}$\{$(\$,\$)$}. .For example,if m=2,n=1,then$L=\{(a^2,a)\}^*,and $ $(a^4,a^2)=(a^2,a)(a^2,a)$ ,so $(a^4,a^2)∈L$. Can you write down the finite automata which recognize $\{(a^1,a^n)\}^*$ ?
The languages described in this question are indeed not regular in general.
Consider the simplest example, $(a,aa)^*$. According to the description of binary languages, this is equivalent to the language $L = \{ (u,v) \mid u = a^n\$^n, v = a^{2n} \}$. But if this language were regular, so would be its projection to the first component, by closure of regular languages under homomorphisms. Thus, $a^n\$^n$ would be regular, contradiction to a well- known result.