  In this paper, we arrive at the notion of equivalence classes of a
non-commutative field exploring some ideas by Soloviev to nonlocal quantum
fields. Specifically, an equivalence relation between non-commutative fields is
formulated by replacing the weak relative locality condition by a weak relative
asymptotic commutativity property, generalizing the notion of relative locality
proved by Borchers in the framework of local QFT. We restrict ourselves to the
simplest case of a scalar field theory with space-space non-commutativity.
