Pinning quasi orders with their endomorphisms
  Some general properties of abstract relations are closely examined. These include generalizations of linearity, and properties based on `pinning' an inequality by a pair of families of endomorphisms.To each property we try to associate a canonical definition of an augmentation (or diminishment) that augments (or diminishes) any given relation to one satisfying the desired property. The motivation behind this study was to identify properties distinguishing between the product ordering and the eventual dominance ordering of the irrationals (the family of functions from N into N), and furthermore to identify their relationship as a member of a natural class of augmentations. 
James Hirschorn