  It is shown that an algebra $\Lambda $ can be lifted with nilpotent Jacobson
radical $r = r(\Lambda)$ and has a generalized matrix unit $\{e_{ii}\}_I$ with
each $\bar e_{ii} $ in the center of $\bar \Lambda = \Lambda /r$ iff $\Lambda $
is isomorphic to a generalized path algebra with weak relations.
Representations of the generalized path algebras are given. As a corollary,
$\Lambda $ is a finite algebra with non-zero unity element over perfect field
$k$ (e.g. a field with characteristic zero or a finite field) iff $\Lambda $ is
isomorphic to a generalized path algebra $k (D, \Omega, \rho)$ of finite
directed graph with weak relations and $dim {\} \Omega < \infty $; $\Lambda $
is a generalized elementary algebra which can be lifted with nilpotent Jacobson
radical and has a complete set of pairwise orthogonal idempotents iff $\Lambda
$ is isomorphic to a path algebra with relations.
