Craig, Andrew; Jäger, Gunther A common framework for lattice-valued uniform spaces and probabilistic uniform limit spaces. (English) Zbl 1184.54006 Fuzzy Sets Syst. 160, No. 9, 1177-1203 (2009). Authors’ summary: We study a category of lattice-valued uniform convergence spaces where the lattice is enriched by two algebraic operations. This general setting allows us to view the category of lattice-valued uniform spaces as a reflective subcategory of our category, and the category of probabilistic uniform limit spaces as a coreflective subcategory. Reviewer: Bernhard Behrens (Göteborg) Cited in 1 ReviewCited in 20 Documents MSC: 54A40 Fuzzy topology 54E15 Uniform structures and generalizations 06F07 Quantales 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 18B30 Categories of topological spaces and continuous mappings (MSC2010) 54B30 Categorical methods in general topology Keywords:category; \(GL\)-monoid; enriched \(cl\)-premonoid; \(L\)-filter; \(L\)-limit space; \(L\)-convergence space; \(L\)-uniform space; \(L\)-uniform convergence space; probabilistic uniform limit space PDFBibTeX XMLCite \textit{A. Craig} and \textit{G. Jäger}, Fuzzy Sets Syst. 160, No. 9, 1177--1203 (2009; Zbl 1184.54006) Full Text: DOI References: [1] Bourbaki, N., General Topology (1990), Springer: Springer Berlin, Heidelberg, New York, London, Paris, Tokyo, (Chapters 1-4) [2] Cook, C. H.; Fischer, H. R., Uniform convergence structures, Math. Ann., 173, 290-306 (1967) · Zbl 0166.18702 [3] Flores, P. V.; Mohapatra, R. N.; Richardson, G., Lattice-valued spaces: fuzzy convergence, Fuzzy Sets Syst., 157, 2706-2714 (2006) · Zbl 1123.54002 [4] Florescu, L. C., Structures syntopogènes probabilistes, Publ. Math. Debrecen, 28, 15-24 (1981) · Zbl 0503.54002 [5] Gutiérrez García, J.; Mardones Pérez, I.; Burton, M. H., The relationship between various filter notions on a GL-monoid, J. Math. Anal. Appl., 230, 291-302 (1999) · Zbl 0916.54006 [6] J. Gutiérrez García, A unified approach to the concept of a fuzzy \(L\); J. Gutiérrez García, A unified approach to the concept of a fuzzy \(L\) [7] Gutiérrez García, J.; Prada Vicente, M. A.; Šostak, A. P., A unified approach to the concept of a fuzzy \(L\)-uniform space, (Rodabaugh, S. E.; Klement, E. P., Topological and Algebraic Structures in Fuzzy Sets (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 81-114 · Zbl 1061.54006 [8] Höhle, U., Probabilistic uniformization of fuzzy uniformities, Fuzzy Sets Syst., 1, 311-332 (1978) · Zbl 0413.54002 [9] Höhle, U., Probabilistic topologies induced by \(L\)-fuzzy uniformities, Manuscripta Math., 38, 289-323 (1982) · Zbl 1004.54500 [10] Höhle, U., Commutative, residuated \(l\)-monoids, (Höhle, U.; Klement, E. P., Nonclassical Logics and their Applications to Fuzzy Subsets (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 53-106 · Zbl 0838.06012 [11] Höhle, U.; Šostak, A. P., Axiomatic foundations of fixed-basis fuzzy topology, (Höhle, U.; Rodabaugh, S. E., Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 123-272 · Zbl 0977.54006 [12] Hutton, B., Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58, 559-571 (1977) · Zbl 0358.54008 [13] Isbell, J. R., Uniform Spaces (1964), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0124.15601 [14] Jäger, G., A category of \(L\)-fuzzy convergence spaces, Quaest. Math., 24, 501-517 (2001) · Zbl 0991.54004 [15] Jäger, G., Subcategories of lattice-valued convergence spaces, Fuzzy Sets Syst., 156, 1-24 (2005) · Zbl 1086.54006 [16] Jäger, G.; Burton, M. H., Stratified \(L\)-uniform convergence spaces, Quaest. Math., 28, 11-36 (2005) · Zbl 1075.54003 [17] Kotzé, W., Uniform spaces, (Höhle, U.; Rodabaugh, S. E., Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, Dordrecht, London), 553-580 · Zbl 0967.54006 [18] Lee, R. S., The category of uniform convergence spaces is cartesian closed, Bull. Austral. Math. Soc., 15, 461-465 (1976) · Zbl 0339.54004 [19] Lowen, R., Fuzzy uniform spaces, J. Math. Anal. Appl., 82, 370-385 (1981) · Zbl 0494.54005 [20] Nusser, H., A generalization of probabilistic uniform spaces, Appl. Categorical Struct., 10, 81-98 (2002) · Zbl 1052.54027 [21] Nusser, H., Completion of probabilistic uniform limit spaces, Quaest. Math., 26, 125-140 (2003) · Zbl 1036.54008 [22] Preuss, G., Theory of Topological Structures (1988), D. Reidel Publishing Company: D. Reidel Publishing Company Dordrecht, Boston, Lancaster, Tokyo [23] Rodabaugh, S. E., Axiomatic foundations for uniform operator quasi-uniformities, (Rodabaugh, S. E.; Klement, E. P., Topological and Algebraic Structures in Fuzzy Sets (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London), 199-234 · Zbl 1041.54012 [24] K.I. Rosenthal, Quantales and their Applications, in: Pitman Research Notes in Mathematics, Vol. 234, Longman, Burnt Mill, Harlow, 1990.; K.I. Rosenthal, Quantales and their Applications, in: Pitman Research Notes in Mathematics, Vol. 234, Longman, Burnt Mill, Harlow, 1990. · Zbl 0703.06007 [25] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland New York · Zbl 0546.60010 [26] Tukey, J. W., Convergence and uniformity in topology (1940), Princeton · Zbl 0025.09102 [27] A.Weil, Sur les éspaces à structure uniforme et sur la topologie générale, Act. Sci. et Ind., Vol. 551, Hermann, Paris, 1937.; A.Weil, Sur les éspaces à structure uniforme et sur la topologie générale, Act. Sci. et Ind., Vol. 551, Hermann, Paris, 1937. · JFM 63.0569.04 [28] O.Wyler, Filter space monads, regularity, completions, in: TOPO 1972—General Topology and its Applications, Lecture Notes in Mathematics, Vol. 378, Springer, Berlin, Heidelberg, New York, 1974, pp. 591-637.; O.Wyler, Filter space monads, regularity, completions, in: TOPO 1972—General Topology and its Applications, Lecture Notes in Mathematics, Vol. 378, Springer, Berlin, Heidelberg, New York, 1974, pp. 591-637. [29] Zhang, D., Stratified Hutton uniform spaces, Fuzzy Sets Syst., 131, 337-346 (2002) · Zbl 1009.54011 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.