Abstract

Quantum B‐algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non‐commutative arithmetic, and quantum theory. A quantum B‐algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B‐module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi‐poset version of the completion of Dedekind and MacNeille.