In Mokris et al. 2014 a novel basis for multivariate hierarchical tensor-product spline spaces was introduced. The construction combines the truncation mechanism (Giannelli et al 2012) with the idea of decoupling basis functions (Mokris et al. 2014). While the former ensures the partition of unity property -essential for geometric modeling applications- the latter allows to obtain a richer set of basis functions compared to previous approaches. Consequently, we can guarantee the completeness property of this novel basis for large classes of multi-level spline spaces. In particular, completeness is obtained for the multi-level spline spaces defined on T-meshes. Most hierarchical spline constructions combine function systems that span certain background spaces with selection procedures. Based on specific assumptions, fulfilled by the background spaces and features of the selection procedures, relevant and useful properties of the resulting hierarchical spline systems, can be obtained. The aim of this study is to extend the approach introduced by Mokris et al. in 2014 to a general abstract framework, that demonstrates how assumptions about the background spaces and features of the selection procedures related to the resulting hierarchical space properties, using an axiomatic approach. This methodology not only unifies (a substantial portion of) the existing theories but also facilitates the development of new constructions. This could open up new possibilities for applications of decoupled hierarchical spline refinements in spline-based numerical simulation via finite and boundary element methods as well as enhancing the flexibility of geometric modeling.