Relation partition algebra — mathematical aspects of uses and part-of relations

Managing complexity in software engineering involves modularisation, grouping design objects into modules, subsystems, etc. This gives rise to new design objects with new ‘use relations’. The lower-level design objects relate to these in a ‘part-of’ relation. But how do ‘use relations’ at different levels of the ‘part-of hierarchy’ relate? We formalise our knowledge on uses and part-of relations, looking for mathematical laws about relations and partitions. A central role is played by an operator /. For a “uses” relation r on a set of objects X and a partitioning into modules viewed as an equivalence θ, we form a relation rθ on the set Xθ. We adopt an axiomatic point of view and investigate a variety of models, corresponding to different abstraction mechanisms and different ways of relating high- and low-level uses relations.