Assume that 
is a field. If 
is a quiver (i.e. a directed graph), the path algebra of 
is denoted by 
. The path algebra has as basis the set of oriented paths of 
(including the stationnary path at each vertex) and the product is induced by the concatenation of the paths.
An admissible ideal of 
is an ideal 
such that 
is generated by linear combinations of paths of length at least 2 and such that 
contains all the paths with length large enough. This technical condition ensures that if 
is an admissible ideal of 
and 
is an admissible ideal of 
and if 
is isomorphic to 
as 
-algebras, then 
and 
are isomorphic as directed graphs.
With these notations, the algebra 
is called presented with the quiver 
and admissible relations given by 
.
For the implementation of the algebra 
, the domain takes a pair theGraph, theRel as parameter, where theGraph encodes the directed graph 
and theRel encodes a list of generating elements of 
.
The oriented graph theGraph is a list with two elements. The first element 
of theGraph is the number of vertices of the graph, we always assume that the vertices are 
. The second element 
of theGraph is a table which assigns to each arrow the list containing the source and the target of this arrow: the elements of this table are of the form alpha=[source of alpha, target of alpha]
.
The list theRel which generates the ideal 
is encoded as follows, an element rel of theRel has the following structure: rel = [ [t_1, path1],..., [t_n, pathn] ] where 
are paths of the graph, i.e. elements of the combinatorial class combinat::graphPaths(theGraph), and 
are scalars. The term rel represents the element 
of the ideal 
.
