  This paper illustrates the combinatorial approach to vertex algebra - study
of vertex algebras presented by generators and relations. A necessary
ingredient of this method is the notion of free vertex algebra. Borcherds
\cite{bor} was the first to note that free vertex algebras do not exist in
general. The reason for this is that vertex algebras do not form a variety of
algebras, because the locality axiom (see sec 2 below) is not an identity.
However, a certain subcategory of vertex algebras, obtained by restricting the
order of locality of generators, has a universal object, which we call the free
vertex algebra corresponding to the given locality bound. In [J. of Algebra,
217(2):496-527] some free vertex algebras were constructed and in certain
special cases their linear bases were found. In this paper we generalize this
construction and find linear bases of an arbitrary free vertex algebra.
  It turns out that free vertex algebras are closely related to the vertex
algebras corresponding to integer lattices. The latter algebras play a very
important role in different areas of mathematics and physics. Here we explore
the relation between free vertex algebras and lattice vertex algebras in much
detail. These results comply with the use of the word "free" in physical
literature refering to some elements of lattice vertex algebras, like in "free
field", "free bozon" or "free fermion".
  Among other things, we find a nice presentation of lattice vertex algebras in
terms of generators and relations, thus giving an alternative construction of
these algebras without using vertex operators. We remark that our construction
works in a very general setting; we do not assume the lattice to be positive
definite, neither non-degenerate, nor of a finite rank.

Quantum Algebra