This paper deals with a particular type 

of generating set, called {\it prefix-free}, 

for a language. 

Given a language $L$ over an alphabet $\Sigma$, 

a set $G$ is a {\it generating set} of $L$, 

denoted by $L \sqsubseteq G$, if $L \subseteq G^+$. 

It is well known that a prefix-free set $G$ has 

the property of {\it unique decipherability} for 

all strings in $G^+$. 



 We first show that for a language $L$, the class 

${\cal PFG}_L$ of all prefix-free and reduced 

generating sets for $L$ is a complete lattice 

under the relation $\sqsubseteq$ and give 

explicitly the least element $G_L^{\inf}$ and the 

greatest element $G_L^{\sup}$. 

Especially we are concerned with the least element 

$G_L^{\inf}$ of the lattice. 

$G_L^{\inf}$ has a {\it good} property in the sense 

that every string in $L$ can be represented by the 

minimum number of strings in $G_L^{\inf}$ among 

${\cal PFG}_L$. 

We give a necessary and sufficient condition for 

$G_L^{\inf}$ to be finite. Moreover, we present 

a polynomial time algorithm for computing $G_L^{\inf}$ 

with respect to the sum of lengths of strings in $L$ 

for a finite language $L$. 

For an infinite language, we consider the problem of 

identifying $G_L^{\inf}$ in the framework of 

{\it identification in the limit} 

proposed by Gold for language learning, and give a 

polynomial time learning algorithm for computing 

$G_L^{\inf}$, provided that the target $G_L^{\inf}$ 

is finite.