Yoneda completeness and flat completeness of ordered fuzzy sets. (English) Zbl 1393.03035
Summary: This paper studies Yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm. Both of these notions are natural extension of directed completeness in order theory to the fuzzy setting. Yoneda completeness requires every forward Cauchy net converges (has a Yoneda limit), while flat completeness requires every flat weight (a counterpart of ideals in partially ordered sets) has a supremum. It is proved that flat completeness implies Yoneda completeness, but, the converse implication holds only in the case that the related triangular norm is either isomorphic to the Łukasiewicz t-norm or to the product t-norm.
Keywords:
category theory; continuous t-norm; quantale; quantaloid; ordered fuzzy set; forward Cauchy net; Yoneda limit; Yoneda complete; flat weight; flat completeReferences:
| [1] | Albert, M. H.; Kelly, G. M., The closure of a class of colimits, J. Pure Appl. Algebra, 51, 1-17 (1988) · Zbl 0656.18004 |
| [2] | America, P.; Rutten, J., Solving reflexive domain equations in a category of complete metric spaces, J. Comput. Syst. Sci., 39, 343-375 (1989) · Zbl 0717.18002 |
| [3] | Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic, 128, 277-298 (2004) · Zbl 1060.03040 |
| [4] | Betti, R.; Carboni, A., Cauchy-completion and the associated sheaf, Cah. Topol. Géom. Différ. Catég., 23, 243-256 (1982) · Zbl 0496.18008 |
| [5] | Birkhoff, G., Lattice Theory, Am. Math. Soc. Coll. Publ., vol. XXV (1973), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0126.03801 |
| [6] | Bonsangue, M. M.; van Breugel, F.; Rutten, J. J.M. M., Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding, Theor. Comput. Sci., 193, 1-51 (1998) · Zbl 0997.54042 |
| [7] | Borceux, F.; Cruciani, R., Skew Ω-sets coincide with Ω-posets, Cah. Topol. Géom. Différ. Catég., 39, 205-220 (1998) · Zbl 0917.18001 |
| [8] | Chen, P.; Lai, H.; Zhang, D., Coreflective hull of finite strong \(L\)-topological spaces, Fuzzy Sets Syst., 182, 79-92 (2011) · Zbl 1250.54008 |
| [9] | Erné, M., Order-topological lattices, Glasg. Math. J., 21, 57-68 (1980) · Zbl 0388.06005 |
| [10] | Flagg, B.; Kopperman, R., Continuity spaces: reconciling domains and metric spaces, Theor. Comput. Sci., 177, 111-138 (1997) · Zbl 0901.68109 |
| [11] | Flagg, R. C.; Sünderhauf, P., The essence of ideal completion in quantitative form, Theor. Comput. Sci., 278, 141-158 (2002) · Zbl 1002.68091 |
| [13] | Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated Lattices, An Algebraic Glimpse at Substructural Logics, Stud. Logic Found. Math., vol. 151 (2007), Elsevier: Elsevier Amsterdam · Zbl 1171.03001 |
| [14] | Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domains, Encycl. Math. Appl., vol. 93 (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1088.06001 |
| [15] | Goubault-Larrecq, J., Non-Hausdorff Topology and Domain Theory, New Math. Monogr., vol. 22 (2013), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1280.54002 |
| [16] | Hájek, P., Metamathematics of Fuzzy Logic, Trends Log., vol. 4 (1998), Springer: Springer Dordrecht · Zbl 0937.03030 |
| [17] | Heymans, H., Sheaves on quantales as generalized metric spaces (2010), Universiteit Antwerpen: Universiteit Antwerpen Belgium, PhD thesis |
| [18] | Hofmann, D.; Reis, C., Probabilistic metric spaces as enriched categories, Fuzzy Sets Syst., 210, 1-21 (2013) · Zbl 1262.18007 |
| [19] | Hofmann, D.; Waszkiewicz, P., A duality of quantale-enriched categories, J. Pure Appl. Algebra, 216, 1866-1878 (2012) · Zbl 1280.18007 |
| [20] | Höhle, U., Many-valued preorders I: the basis of many-valued mathematics, (Magdalena, L.; Verdagay, J. L.; Esteva, F., Enric Trillas: A Passion for Fuzzy Sets. A Collection of recent Works on Fuzzy Logic. Enric Trillas: A Passion for Fuzzy Sets. A Collection of recent Works on Fuzzy Logic, Stud. Fuzziness Soft Comput., vol. 322 (2015), Springer), 125-150 · Zbl 1383.06001 |
| [21] | Höhle, U.; Kubiak, T., A non-commutative and non-idempotent theory of quantale sets, Fuzzy Sets Syst., 166, 1-43 (2011) · Zbl 1226.06011 |
| [22] | Kelly, G. M., Basic Concepts of Enriched Category Theory, Lond. Math. Soc. Lect. Note Ser., vol. 64 (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0478.18005 |
| [23] | Kelly, G. M.; Schmitt, V., Notes on enriched categories with colimits of some class, Theory Appl. Categ., 14, 399-423 (2005) · Zbl 1082.18004 |
| [24] | Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms, Trends Log., vol. 8 (2000), Springer: Springer Dordrecht · Zbl 0972.03002 |
| [25] | Kostanek, M.; Waszkiewicz, P., The formal ball model for \(Q\)-categories, Math. Struct. Comput. Sci., 21, 1, 41-64 (2011) · Zbl 1215.18005 |
| [26] | Künzi, H.; Schellekens, M., On the Yoneda completion of a quasi-metric space, Theor. Comput. Sci., 278, 159-194 (2002) · Zbl 1025.54014 |
| [27] | Künzi, H.-P.; Pajoohesh, H.; Schellekens, M., Partial quasi-metrics, Theor. Comput. Sci., 365, 237-246 (2006) · Zbl 1109.54021 |
| [28] | Lai, H.; Zhang, D., On the embedding of Top in the category of stratified \(L\)-topological spaces, Chin. Ann. Math., 26, 219-228 (2005) · Zbl 1075.54004 |
| [29] | Lai, H.; Zhang, D., Fuzzy preorder and fuzzy topology, Fuzzy Sets Syst., 157, 1865-1885 (2006) · Zbl 1118.54008 |
| [30] | Lai, H.; Zhang, D., Complete and directed complete Ω-categories, Theor. Comput. Sci., 388, 1-25 (2007) · Zbl 1131.18006 |
| [31] | Lai, H.; Zhang, D., Closedness of the category of liminf complete fuzzy orders, Fuzzy Sets Syst., 282, 86-98 (2016) · Zbl 1392.18004 |
| [32] | Lawvere, F. W., Metric spaces, generalized logic and closed categories, Rend. Semin. Mat. Fis. Milano, XLIII, 135-166 (1973) · Zbl 0335.18006 |
| [33] | Matthews, S. G., Partial metric topology, Ann. N.Y. Acad. Sci., 728, 183-197 (1994) · Zbl 0911.54025 |
| [34] | Mostert, P. S.; Shields, A. L., On the structure of semigroups on a compact manifold with boundary, Ann. Math., 65, 117-143 (1957) · Zbl 0096.01203 |
| [35] | Pu, Q.; Zhang, D., Preordered sets valued in a GL-monoid, Fuzzy Sets Syst., 187, 1-32 (2012) · Zbl 1262.18008 |
| [36] | Pu, Q.; Zhang, D., A note on divisible residuated lattices, J. Sichuan Univ. (Natural Science Edition), 49, 985-989 (2012) · Zbl 1274.06054 |
| [37] | Rosenthal, K. I., Quantales and Their Applications, Pitman Res. Notes Math. Ser., vol. 234 (1990), Longman: Longman Harlow · Zbl 0703.06007 |
| [38] | Rosenthal, K. I., The Theory of Quantaloids, Pitman Res. Notes Math. Ser., vol. 348 (1996), Longman: Longman Harlow · Zbl 0845.18003 |
| [39] | Rutten, J. J.M. M., Weighted colimits and formal balls in generalized metric spaces, Topol. Appl., 89, 179-202 (1998) · Zbl 0982.54029 |
| [40] | Schmitt, V., Completions of non-symmetric metric spaces via enriched categories, Georgian Math. J., 16, 157-182 (2009) · Zbl 1173.18004 |
| [41] | Stubbe, I., Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory Appl. Categ., 14, 1-45 (2005) · Zbl 1079.18005 |
| [42] | Stubbe, I., An introduction to quantaloid-enriched categories, Fuzzy Sets Syst., 256, 95-116 (2014) · Zbl 1335.18002 |
| [43] | Tao, Y.; Lai, H.; Zhang, D., Quantale-valued preorders: globalization and cocompleteness, Fuzzy Sets Syst., 256, 236-251 (2014) · Zbl 1337.06010 |
| [44] | Vickers, S., Localic completion of generalized metric spaces. I, Theory Appl. Categ., 14, 328-356 (2005) · Zbl 1083.54019 |
| [45] | Wagner, K. R., Solving recursive domain equations with enriched categories (1994), Carnegie Mellon University: Carnegie Mellon University Pittsburgh, PhD thesis |
| [46] | Wagner, K. R., Liminf convergence in Ω-categories, Theor. Comput. Sci., 184, 61-104 (1997) · Zbl 0935.18008 |
| [47] | Walters, R. F.C., Sheaves and Cauchy-complete categories, Cah. Topol. Géom. Différ. Catég., 22, 283-286 (1981) · Zbl 0495.18009 |
| [48] | Waszkiewicz, P., On domain theory over Girard quantales, Fundam. Inform., 92, 169-192 (2009) · Zbl 1193.06006 |
| [49] | Zadeh, L. A., Similarity relations and fuzzy orderings, Inf. Sci., 3, 177-200 (1971) · Zbl 0218.02058 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
