Formal balls of Q-categories | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 2، دوره 18، شماره 1، اردیبهشت 2023، صفحه 1-18 اصل مقاله (508.94 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.52547/cgasa.2022.102914 | ||
| نویسندگان | ||
| Xianbo Yang؛ Dexue Zhang | ||
| School of Mathematics, Sichuan University, Chengdu, China | ||
| چکیده | ||
| The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized to Q-categories by Kostanek and Waszkiewicz, where Q is a commutative and unital quantale. This paper concerns the influence of the structure of the quantale Q on the connection between Yoneda completeness of Q-categories and directed completeness of their sets of formal balls. In the case that Q is the unit interval [0, 1] equipped with a continuous t-norm &, it is shown that in order that Yoneda completeness of each Q-category be equivalent to directed completeness of its set of formal balls, a necessary and sufficient condition is that the t-norm & is Archimedean. | ||
| کلیدواژهها | ||
| Q-category؛ formal ball؛ continuous t-norm؛ quantale | ||
| مراجع | ||
|
[1] Ali-Akbari, M., Honari, B., Pourmahdian, M., and Rezaii, M. M., The space of formal balls and models of quasi-metric spaces, Math. Struct. Comput. Sci. 19 (2009), 337- 355.
[2] Edalat, A. and Heckmann, R., A computational model for metric spaces, Theor. Comput. Sci. 193 (1998), 53-73.
[3] Flagg, R.C., Sünderhauf, P., and Wagner, K.R., A logical approach to quantitative domain theory, Topology Atlas, Preprint 23, 1996.
[4] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., “Continuous Lattices and Domains”, Cambridge University Press, 2003.
[5] Goubault-Larrecq, J., “Non-Hausdorff Topology and Domain Theory”, Cambridge University Press, 2013.
[6] Goubault-Larrecq, J., Formal ball monads, Topol. Appl. 263 (2019), 372-391.
[7] Goubault-Larrecq, J. and Ng, K.M., A few notes on formal balls, Log. Methods Comput. Sci. 13(4:18) (2017), 1-34.
[8] Hofmann, D., Seal, G.J., and Tholen, W., “Monoidal Topology: A Categorical Approach to Order, Metric, and Topology”, Cambridge University Press, 2014.
[9] Klement, E.P., Mesiar, R. and Pap, E., “Triangular Norms”, Kluwer Academic Publishers, 2000.
[10] Kostanek, M. and Waszkiewicz, P., The formal ball model for Q-categories, Math. Struct. Comput. Sci. 21 (2011), 41-64.
[11] Künzi, H.P. and Schellekens, M.P., On the Yoneda completion of a quasi-metric space, Theor. Comput. Sci. 278 (2002), 159-194.
[12] Lai, H. and Zhang, D., Complete and directed complete Ω-categories, Theor. Comput. Sci. 388 (2007), 1-25.
[13] Lai, H., Zhang, D., and Zhang, G., A comparative study of ideals in fuzzy orders, Fuzzy Sets Syst. 382 (2020) 1-28.
[14] Lawvere, F.W., Metric spaces, generalized logic, and closed categories, Rend. Semin. Mat. Fis. Milano 43 (1973), 135-166.
[15] Li, W. and Zhang, D., Sober metric approach spaces, Topol. Appl. 233 (2018), 67-88.
[16] Romaguera, S. and Valero, O., Domain theoretic characterisations of quasi-metric completeness in terms of formal balls, Math. Struct. Comput. Sci. 20 (2010), 453- 472.
[17] Rosenthal, K.I., “Quantales and Their Applications”, Longman, 1990.
[18] Rutten, J.J.M.M., Weighted colimits and formal balls in generalized metric spaces, Topol. Appl. 89 (1998), 179-202.
[19] Smyth, M.B., Quasi-uniformities: Reconciling domains with metric spaces, in: “Mathematical Foundations of Programming Language Semantics”, Lecture Notes in Computer Science, Vol. 298, pp. 236-253, Springer, 1988.
[20] Vickers, S., Localic completion of generalized metric spaces, Theory Appl. Categ. 14 (2005), 328-356.
[21] Wagner, K.R., Liminf convergence in Ω-categories, Theor. Comput. Sci. 184 (1997), 61-104.
[22] Waszkiewicz, P., On domain theory over Girard quantales, Fund. Infor. 92 (2009), 169-192.
[23] Weihrauch, K. and Schreiber, U., Embedding metric spaces into cpo’s, Theor. Com- put. Sci. 16 (1981), 5-24. | ||
|
آمار تعداد مشاهده مقاله: 251 تعداد دریافت فایل اصل مقاله: 854 |
||