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enter image description here Questions:

  1. Give an example of a homomorphism, using the same alphabet, Σ, for both languages A and B.
  2. Now, give a second example of a homomorphism but this time using two different alphabets, Σ and Γ, for languages A and B, respectively.

Questions to stack overflow:

  • How do I give the examples above? Do I make the state diagrams? Do I show it through the tuple-definition?
  • What are some examples of the questions above?
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    $\begingroup$ A homomorphism $f$ in this context maps from characters of $\Sigma$ to strings of $\Gamma^*$. So to define a specific $f$ you can just give a mapping for each character from $\Sigma$. $\endgroup$ – siracusa Oct 19 at 4:32
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Notice the abuse of notation used. The quote overload symbol $f$ with three meanings which I will now denote differently with $f,f',f''$. First is map from alphabet to words $$f: \Sigma \rightarrow \Gamma^*$$. The second is the map between words $$f':\Sigma^* \rightarrow \Gamma^*$$. The third is the map between languages $$f'':\mathbb{P}(\Sigma^*) \rightarrow \mathbb{P}(\Gamma^*)$$.

The The main point of the quote is that given $f$ there is the particular way to construct the corresponding $f'$ and $f''$. So, what one might call homomorphisms between languages $f''$ is specified by just giving map from alphabet to words $f$. And $f$ is usually easier to describe directly than $f''$.

Another important observation is that not all function of type $$\mathbb{P}(\Sigma^*) \rightarrow \mathbb{P}(\Gamma^*)$$ is a homomorphism of languages. (Because it doesn't "preserve structure" w.r.t. concatenation operation languages are equipped with.) However, the said construction is special in that given any $f$ with the correct type, $f''$ will be a valid homomorphism of languages.

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