Editing Free monoid (section)

==Endomorphisms==
An '''[[endomorphism]]''' of ''A''<sup>&lowast;</sup> is a morphism from ''A''<sup>&lowast;</sup> to itself.<ref name=LotII450>{{harvtxt|Lothaire|2011|p=450}}</ref>  The [[identity map]] ''I'' is an endomorphism of ''A''<sup>&lowast;</sup>, and the endomorphisms form a [[monoid]] under [[composition of functions]].

An endomorphism ''f'' is '''prolongable''' if there is a letter ''a'' such that ''f''(''a'') = ''as'' for a non-empty string ''s''.<ref name=AS10>Allouche & Shallit (2003) p.10</ref>

===String projection===
The operation of [[String operations#String projection|string projection]] is an endomorphism. That is, given a letter ''a'' &isin; &Sigma; and a string ''s'' &isin; &Sigma;<sup>&lowast;</sup>, the string projection ''p''<sub>a</sub>(''s'') removes every occurrence of ''a'' from ''s''; it is formally defined by

:<math>p_a(s) = \begin{cases} 
\varepsilon & \text{if } s=\varepsilon, \text{ the empty string} \\
p_a(t) & \text{if } s=ta  \\ 
p_a(t)b & \text{if } s=tb \text{ and } b\ne a.   
\end{cases}</math>

Note that string projection is well-defined even if the rank of the monoid is infinite, as the above recursive definition works for all strings of finite length. String projection is a [[morphism]] in the category of free monoids, so that

:<math>p_a\left(\Sigma^*\right)= \left(\Sigma-a\right)^*</math>

where <math>p_a\left(\Sigma^*\right)</math> is understood to be the free monoid of all finite strings that don't contain the letter ''a''.  Projection commutes with the operation of string concatenation, so that <math>p_a(st)=p_a(s)p_a(t)</math> for all strings ''s'' and ''t''. There are many right inverses to string projection, and thus it is a [[split epimorphism]].

The identity morphism is <math>p_\varepsilon,</math> defined as <math>p_\varepsilon(s)=s</math> for all strings ''s'', and <math>p_\varepsilon(\varepsilon)=\varepsilon</math>. 

String projection is commutative, as clearly

:<math>p_a(p_b(s))=p_b(p_a(s)).</math>

For free monoids of finite rank, this follows from the fact that free monoids of the same rank are isomorphic, as projection reduces the rank of the monoid by one.

String projection is [[idempotent]], as

:<math>p_a(p_a(s))=p_a(s)</math>

for all strings ''s''. Thus, projection is an idempotent, commutative operation, and so it forms a bounded [[semilattice]] or a commutative [[band (algebra)|band]].