Leanest quasi-orderings

A convenient method for defining a quasi-ordering, such as those used for proving termination of rewriting, is to choose the minimum of a set of quasi-orderings satisfying some desired traits. Unfortunately, a minimum in terms of set inclusion can be non-existent even when an intuitive “minimum” exists. We suggest an alternative to set inclusion, called “leanness”, show that leanness is a partial order on quasi-orderings, and provide sufficient conditions for the existence of a “leanest” member of a set of total well-founded quasi-orderings.