restricted homomorphism
Let $h$ be a homomorphism^{} over an alphabet $\mathrm{\Sigma}$. Let $L$ be a language^{} over $\mathrm{\Sigma}$. We say that $h$ is $k$restricted on $L$ if

1.
there is a letter $b\in \mathrm{Alpha}(L)$ such that no word in $L$ begins with $b$ and contains more than $k1$ consecutive occurrences of $b$ in it,

2.
for any $a\in \mathrm{Alpha}(L)$,
$$h(a)=\{\begin{array}{cc}\lambda \hfill & \text{if}a=b\hfill \\ a\hfill & \text{otherwise.}\hfill \end{array}$$
Here, $\mathrm{Alpha}(L)$ is the set of all letters in $\mathrm{\Sigma}$ that occur in words of $L$.
It is easy to see that any $k$restricted homomorphism on $L$ is a $k$linear erasing on $L$, for if $u\in L$ is a nonempty word, then we may write $u={v}_{1}{b}^{{m}_{1}}{v}_{2}{b}^{{m}_{2}}\mathrm{\cdots}{v}_{n}{b}^{{m}_{n}}$, where each $$, and each ${v}_{i}$ is a nonempty word not containing any occurrences of $b$. Then
$$u={v}_{1}\mathrm{\cdots}{v}_{n1}+\sum _{i=1}^{n}{m}_{i}\le h(u)+n(k1)\le h(u)+(k1)h(u)=kh(u).$$ 
Note that $n\le h(u)$ since $1\le {v}_{i}$ for each $i=1,\mathrm{\dots},n$. A $k$linear erasing is in general not a $k$restricted homomorphism, an example of which is the following: $L={\{a,ab\}}^{*}$ and $h:\{a,b\}\to \{a,b\}$ given by $h(a)={a}^{2}$ and $h(b)=\lambda $. Then $h$ is a $1$linear erasing, but not a $1$restricted homomorphism, on $L$.
A family $\mathcal{F}$ of languages is said to be closed under^{} restricted homomorphism if for every $L\in \mathcal{F}$, and every $k$restricted homomorphism $h$ on $L$, $h(L)\in \mathcal{F}$. By the previous paragraph, we see that if $\mathcal{F}$ is closed under linear erasing, it is closed under restricted homomorphism. The converse^{} of this is not necessarily true.
References
 1 A. Salomaa, Formal Languages^{}, Academic Press, New York (1973).
Title  restricted homomorphism 

Canonical name  RestrictedHomomorphism 
Date of creation  20130322 18:59:11 
Last modified on  20130322 18:59:11 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 68Q45 
Synonym  $k$restricted homomorphism 
Synonym  krestricted homomorphism 
Related topic  LinearErasing 