  In this paper we build an abstract description of vertex algebras from their
basic axioms. Starting with Borcherds' notion of a vertex group, we naturally
construct a family of multilinear singular maps parameterised by trees. These
singular maps are defined in a way which focusses on the relations of
singularities to their inputs. In particular we show that this description of a
vertex algebra allows us to present generalised notions of rationality,
commutativity and associativity as natural consequences of the definition.
Finally, we show that for a certain choice of vertex group, axiomatic vertex
algebras correspond bijectively to algebras in the relaxed multilinear category
of representations of a vertex group.
