I'm not sure if I get the following definition right:1.Def: A map $\varphi: \Sigma^{*} \rightarrow 2^{\Delta^{*}}$ is called Substitution iff:$\forall u,v \in \Sigma^{*}: \varphi(uv)=\varphi(u) \varphi(v)$$\varphi(\epsilon)=\{\epsilon\}$My assumption: every string in $\Sigma^{*}$ with the size $>0$ maps to a Language from $\Delta^{*}$.such as:$\Sigma := \{a, b\}$, $\Delta:=\{0,1\}$$\varphi(a) = \{01^{i}0 |i \in \mathbb N \}$, $\varphi(b) = \{w\in \Delta^{*} ||w|_{0} \neq |w|_{1}\}$$\varphi(ab) = \varphi(a)\varphi(b)=\{01^{i}0w |i \in \mathbb N \land w \in \Delta^{*} |w|_{0} \neq |w|_{1}\}$example strings: 001, 000, 0101, 01011, 010100, ...My first question: Is this assumption and example right?2.Def: $\varphi$ is finitie iff $\varphi(a)$ $\forall a \in \Sigma$ is a finite subset of $\Delta^{*}$My assumption: $\Delta^{*}$ is every possible string over the alphabet $\Delta$ and an finite subset would be a limited set $X \subseteq \Delta^{*}$  $|X| < \infty$I would now implie that my first example is not a finite substitution, because a and b are not mapping to finite sets.an example could be:$\varphi(a) = \{10^{i}1 | i \in \{1,...,100\}\}$, $\varphi(b) = \{\epsilon\}$$\varphi(bba) = \varphi(b)\varphi(b)\varphi(a) = \{\epsilon\}\{\epsilon\}\{10^{i}1 | i \in \{1,...,100\}\}=\{10^{i}1 | i \in \{1,...,100\}\}$now I can implie that the class $REG$ is closed under finite substitution, because a concatenation of finite sets is finite and every finite set is regular.My second question: Is this my assumption about finite substitution and the closure propierty right?
Finite substitution and regular closure properties
formal languages;regular languages;closure properties
You seem to have understood the operation of substitution. That maps any string into a language by concatenating the images of the letters. Your argument that regular languages are closed under finite substitution does not work. Even if the substitution maps avery word into a finite language, the substitution of a regular language usually is infinite as the initial ragular language is infinite.A possible approach is to use regular expressions. 