Prove irregularity of a language using closure properties

Question

Given the language $L=\{a^{j+1}b^kc^{j-k}|j\ge k\ge 0 \}$ I need to prove that it is not a regular language using closure properties.

I was having a trouble handling $a^{j+1}$ so I tried to prove this first for $L_1 = \{a^jb^kc^{j-k}|j\ge k\ge 0 \}$, I assume $L_1$ is regular and define an homomorphism $h: \{a,b,c\}\rightarrow \{a,b\}$ such that $$h(a) = b,\space h(b) = a,\space h(c)=\varepsilon$$ and I get $L_2=h(L_1)= \{b^ja^k|j\ge k\ge 0 \}$ which is also regular by my assumption and the closure of homomorphism, and from the reverse closure I get $L_3 = L_2^R = \{a^kb^j|j\ge k\ge 0 \}$ is also a regular language but I already know that $L_3$ isn't regular, a contradiction, therefore, $L_1$ isn't regular.

Going back to the original language $L$, I think I need to find a way to reach $L_1$ from $L$ while using closure properties so the assumption of $L$'s regularity leads to a contradiction, but so far I couldn't find any

Answer 1

Suppose that $L$ is regular. Since regular languages are closed under intersection, then $$ K = L \cap a^*c^* = \{a^{j+1}c^j \mid j \geqslant 0 \} $$ would be regular and the left quotient of $K$ by $a$ $$ a^{-1}K = \{u \mid au \in K\} = \{a^jc^j \mid j \geqslant 0 \} $$ would also be regular. I suppose you already know that this latter language is not regular. Thus $L$ is not regular.

Answer 2

Assume $L_3 = \{a^kb^j|j\ge k\ge 0 \}$ is the only language that is known to be non-regular.

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Let $L_4=\{b\}$, the language that contains one string, $c$. Then $(h(L\circ L_4)\cup\{\epsilon\})^R=L_3$. Since $L_3$ is non_regular but $ L_4$ and $\{\epsilon\}$ are regular and regular languages are closed under concatenation, homomorphism, union and reversal, $L$ must be non-regular.

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Let $L_5=\{a\}$. Then $(h(L_5\backslash L))^R=L_3$, where $L_5\backslash L$ is the left quotient of $L$ by $L_5$. Since $L_5$ is regular and regular languages are closed under left quotient, homomorphism and reversal, $L$ must be non-regular.

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It might be more reasonable to assume that $L_6=\{a^nb^n\mid n\ge0\}$ is known to be non-regular, since that is the first language presented as non-regular in many textbooks.

Exercise. With the assumption above, show $L$ is non-regular using closure properties.