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# [Solved]: Concatenation property of regular languages

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Problem Detail:

If L is the empty set and therefore a regular language, I know that L concatenated with sigma star is equal L; Are there any other languages that, when concatenated with sigma star will result in the same language?

\$L \Sigma^* = \emptyset \Sigma^* = \emptyset = L\$

\$\Sigma^*L = \Sigma^* \emptyset = \emptyset = L\$

\$L \Sigma^* = (S \Sigma^*)\Sigma^* = S \Sigma^* = L\$ for all languages \$S\$

\$\Sigma^* L = \Sigma^* (\Sigma^* S) = \Sigma^* S = L\$ for all languages S

\$L \Sigma^* = (\Sigma^* S \Sigma^*) \Sigma^* = \Sigma^* S \Sigma^* = L\$ for all languages \$S\$

\$\Sigma^* L = \Sigma^* (\Sigma^* S \Sigma^*) = \Sigma^* S \Sigma^* = L\$ for all languages S.

In short: there are infinitely many distinct languages where appending the language to \$\Sigma^*\$ (front or back) will yield the same language. An infinite family of such languages is given by \$\Sigma^* S \Sigma^*\$ where \$S\$ can be any language whatsoever.