  We obtain a complete structural characterization of Cohn-Leavitt algebras
over no-exit objects as graded involutive algebras. Corollaries of this result
include graph-theoretic conditions characterizing when a Leavitt path algebra
is a directed union of (graded) matricial algebras over the underlying field
and over the algebra of Laurent polynomials and when the monoid of isomorphism
classes of finitely generated projective modules is atomic and cancellative.
  We introduce the non-unital generalizations of graded analogues of noetherian
and artinian rings, graded locally noetherian and graded locally artinian
rings, and characterize graded locally noetherian and graded locally artinian
Leavitt path algebras without any restriction on the cardinality of the graph.
As a consequence, we relax the assumptions of the Abrams-Aranda-Perera-Siles
characterization of locally noetherian and locally artinian Leavitt path
algebras.
