%0 Journal Article
%T Separated finitely supported $Cb$-sets
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Keshvardoost, Khadijeh
%A Mahmoudi, Mojgan
%D 2020
%\ 07/01/2020
%V 13
%N 1
%P 55-82
%! Separated finitely supported $Cb$-sets
%K Finitely supported $Cb$-sets
%K nominal set
%K $S$-set
%K support
%K simple
%R
%X 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.
%U http://cgasa.sbu.ac.ir/article_87413_0b1bd3b91cc24e487df84a5e89e77f28.pdf