The notion of information systems provides a convenient tool for knowledge representation of objects in terms of their attribute values, while partial ordering is usually used to research the rough monotonicity of an uncertainty measure in information systems. In this paper, we first reveal the limitations of existing partial orderings to describe information granulations in information systems with several illustrative examples. Then, a generalized partial ordering with a set-size nature is proposed to overcome their shortcoming and some of its important properties are derived. Finally, we prove that several existing information granulations all satisfy the granulation monotonicity induced by the proposed partial ordering. The presented partial ordering appears to be well suited to characterize the nature of information granulations in an information system. These results will be very helpful for studying granular computing and uncertainty in information systems.