Separated finitely supported $Cb$-sets | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 5، دوره 13، شماره 1، مهر 2020، صفحه 55-82 اصل مقاله (525.72 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.13.1.55 | ||
| نویسندگان | ||
| Khadijeh Keshvardoost1؛ Mojgan Mahmoudi2 | ||
| 1Department of Mathematics, Velayat University, Iranshahr, Sistan and Baluchestan, Iran. | ||
| 2Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Finitely supported $Cb$-sets؛ nominal set؛ $S$-set؛ support؛ simple | ||
| مراجع | ||
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[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories", John Wiley and Sons, 1990. [2] Burris, S. and Sankappanavar, H.P., "A Course in Universal Algebra", Springer- Velag, 1981. [3] Ebrahimi, M.M., Keshvardoost, Kh., and Mahmoudi, M., Simple and subdirectly irreducible finitely supported Cb-sets, Theort. Comput. Sci. 706 (2018), 1-21. [4] Ebrahimi, M.M. and Mahmoudi, M., The category of M-sets, Ital. J. Pure Appl. Math. 9 (2001), 123-132. [5] Gabbay, M. and Pitts, A., A new approach to abstract syntax with variable binding, Form. Asp. Comput. 13(3-5) (2002), 341-363. [6] Herrlich, H. and Strecker, G., Coreflective subcategories, Trans. Amer. Math. Soc. 157 (1971), 205-226. [7] Kilp, M., Knauer, U. and Mikhalev, A., "Monoids, Acts and Categories", Walter de Gruyter, 2000. [8] Pitts, A., "Nominal sets, Names and Symmetry in Computer Science", Cambridge University Press, 2013. [9] Pitts, A., Nominal presentations of the cubical sets model of type theory, LIPIcs. Leibniz Int. Proc. Inform. (2015), 202-220. | ||
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