# A map from one set of words to another that is not a morphism

A word is a concatenation of letters from a non-empty set called an alphabet. For example, if the alphabet is $\{a,b\}$, then $bba$ is a word from that alphabet.

Let the set of all finite words made up of letters chosen from an alphabet $\Sigma$ be denoted by $\Sigma^*$.

Let $\Sigma$ and $\Delta$ be alphabets. A morphism is a map $h$ from a $\Sigma^*$ to $\Delta^*$ that obeys the identity $h(xy)=h(x)h(y)$ for all words $x,y$ from $\Sigma^*$.

For example, given an alphabet $\Sigma=\{a,b\}$, an example of a morphism $h$ from $\Sigma^*$ to $\Sigma^*$ is $h(a)=ab$ and $h(b)=b$. Thus, $h(ab)=h(a)h(b)=abb$.

Is there a name for a map from $\Sigma^*$ to $\Delta^*$ that is not a morphism?

For example, given an alphabet $\Sigma=\{a,b\}$, what is $g$ called if part of its definition includes $g(a)=ab$, $g(b)=b$, and $g(ab)=aa$?

• There is no reason to give a name to such a negative property. Besides, an alphabet might be empty. – J.-E. Pin Jul 9 '14 at 12:01