Axiomatic Gi-vertex algebras

Inspired by Borcherds' work on "G-vertex algebras," we formulate and study an axiomatic counterpart of Borcherds' notion of G-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by G(1). Specifically, we formulate a notion of axiomatic G(1)-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic G(1)-vertex algebra is a nonlocal generalization of the notion of vertex-algebra. We also show how to construct axiomatic G(1)-vertex algebras from a set of compatible G(1)-vertex operators.