  Let $K\ < x_i\ >$ be the free associative algebra generated by a finite or
countable number of variables $x_i$. The notion of "letterplace correspondence"
introduced in [1,2] for the graded (two-sided) ideals of $K\ < x_i\ >$ is
extended in this paper also to the nongraded case. This amounts to the
possibility of modelizing nongraded noncommutative presented algebras by means
of a class of graded commutative algebras that are invariant under the action
of the monoid $\mathbb N$ of natural numbers. For such purpose we develop the
notion of saturation for the graded ideals of $K\ < x_i,t\ >$, where $t$ is an
extra variable and for their letterplace analogues in the commutative
polynomial algebra $K[x_{ij},t_j]$, where $j$ ranges in $\mathbb N$. In
particular, one obtains an alternative algorithm for computing inhomogeneous
noncommutative Gr\"obner bases using just homogeneous commutative polynomials.
The feasibility of the proposed methods is shown by an experimental
implementation developed in the computer algebra system Maple and by using
standard routines for the Buchberger algorithm contained in Singular.
  References
  [1] La Scala, R.; Levandovskyy, V., Letterplace ideals and non-commutative
Gr\"obner bases. J. Symbolic Comput., 44 (2009), 1374--1393.
  [2] La Scala, R.; Levandovskyy, V., Skew polynomial rings, Gr\"obner bases
and the letterplace embedding of the free associative algebra. J. Symbolic
Comput., 48 (2013), 110--131
