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Abstract: Invariant assertions play an important role in the analysis and documentation of while loops of imperative programs. Invariant functions and invariant relations are alternative analysis tools that are distinct from invariant assertions but are related to them. In this paper we discuss these three concepts and analyze their relationships. The study of invariant functions and invariant relations is interesting not only because it provides alternative means to analyze loops, but also because it gives us insights into the structure of invariant assertions, hence it may help us enhance techniques for generating invariant assertions. [Copyright &y& Elsevier]