separated finitely supported cb-sets

The monoid cb of name substitutions and the notion of finitelysupported cb-sets introduced by pitts as a generalization of nominal sets. asimple finitely supported cb-set is a one point extension of a cyclic nominalset. the support map of a simple finitely supported cb-set is an injectivemap. also, for every two distinct elements of a simple finitely supportedcb-set, there exists an element of the monoid cb which separates them bymaking just one of them into an element with the empty support.in this paper, we generalize these properties of simple finitely supportedcb-sets by modifying slightly the notion of the support map; defining the notionof 2-equivariant support map; and introducing the notions of s-separatedand z-separated finitely supported cb-sets. we show that the notions of sseparatedand z-separated coincide for a finitely supported cb-set whose supportmap is 2-equivariant. among other results, we find a characterizationof simple s-separated (or z-separated) finitely supported cb-sets. finally,we show that some subcategories of finitely supported cb-sets with injectiveequivariant maps which constructed applying the defined notions are reflective.