The over operator is commonly used for α-blending in various visualization techniques. In the current form, it is a binary operator and must respect the restriction of order dependency, hence posing a significant performance limit. This paper proposes a fully generalized version of this operator. Compared with its predecessor, the fully generalized over operator is not only n-operator compatible but also any-order friendly. To demonstrate the advantages of the proposed operator, we apply it to the asynchronous and order-dependent image composition problem in parallel visualization for big data science and further parallelize it for performance improvement. We conduct theoretical analyses to establish the performance superiority of the proposed over operator in comparison with its original form, which is further validated by extensive experimental results in the context of real-life scientific visualization.