Separated finitely supported $Cb$-sets

Document Type : Research Paper


1 Department of Mathematics, Velayat University, Iranshahr, Sistan and Baluchestan, Iran.

2 Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran.



The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists an element of the monoid $Cb$ which separates them by making just one of them into an element with the empty support.
In this paper, we generalize these properties of simple finitely supported $Cb$-sets by modifying slightly the notion of the support map; defining the notion of $\mathsf{2}$-equivariant support map; and introducing the notions of s-separated and z-separated finitely supported $Cb$-sets. We show that the notions of s-separated and z-separated coincide for a finitely supported $Cb$-set whose support map is $\mathsf{2}$-equivariant. Among other results, we find a characterization of simple s-separated (or z-separated) finitely supported $Cb$-sets. Finally, we show that some subcategories of finitely supported $Cb$-sets with injective equivariant maps which constructed applying the defined notions are reflective.


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