free submonoid
Let $A$ be an arbitrary set, let ${A}^{\ast}$ be the free monoid on $A$, and let $e$ be the identity element^{} (empty word^{}) of ${A}^{\ast}$.
Let $M$ be a submonoid of ${A}^{\ast}$. The minimal generating set of $M$ is
$$\mathrm{mgs}(M)=(M\setminus \{e\})\setminus {(M\setminus \{e\})}^{2}.$$  (1) 
Shortly, $\mathrm{mgs}(M)$ is the set of all the nontrivial elements of $M$ that cannot be “reconstructed” as products of elements of $M$. It is straightforward that

1.
${(\mathrm{mgs}(M))}^{\ast}=M$, and

2.
if $S\subseteq {A}^{\ast}$ and $M\subseteq {S}^{\ast}$, then $\mathrm{mgs}(M)\subseteq S$.
We say that $M$ is a free submonoid of ${A}^{\ast}$ if it is isomorphic^{} (as a monoid) to a free monoid ${B}^{\ast}$ for some set $B$. A set $K\subseteq {A}^{\ast}$ such that $K=\mathrm{mgs}(M)$ for some free submonoid $M$ of ${A}^{\ast}$ is also called a code.
Title  free submonoid 

Canonical name  FreeSubmonoid 
Date of creation  20130322 18:21:36 
Last modified on  20130322 18:21:36 
Owner  Ziosilvio (18733) 
Last modified by  Ziosilvio (18733) 
Numerical id  5 
Author  Ziosilvio (18733) 
Entry type  Definition 
Classification  msc 20M10 
Classification  msc 20M05 
Defines  minimal generating set of a submonoid 