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Keywords:
many-value topology; monadic category; nucleus; quantale; quantale algebra; quantale algebroid; quantale module; quantaloid; tensor product
many-value topology; monadic category; nucleus; quantale; quantale algebra; quantale algebroid; quantale module; quantaloid; tensor product
Summary:
The paper considers a fuzzification of the notion of quantaloid of K. I. Rosenthal, which replaces enrichment in the category of $\bigvee$-semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At the end of the paper, we prove that the category of quantale algebroids has a monoidal structure given by tensor product.
References:
[1] Abramsky, S., Vickers, S.: Quantales, observational logic and process semantics. Math. Struct. Comput. Sci. 3 (1993), 161–227. DOI 10.1017/S0960129500000189 | MR 1224222 | Zbl 0823.06011
[2] Adámek, J., Herrlich, H., Strecker, G. E.: Abstract and Concrete Categories: The Joy of Cats. Dover Publications, Mineola, New York 2009. MR 1051419
[3] Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules. Second edition. Springer-Verlag, 1992. MR 1245487 | Zbl 0765.16001
[4] Betti, R., Kasangian, S.: Tree automata and enriched category theory. Rend. Ist. Mat. Univ. Trieste 17 (1985), 71–78. MR 0863557 | Zbl 0614.68045
[5] Borceux, F.: Handbook of Categorical Algebra. Volume 2: Categories and Structures. Cambridge University Press, 1994. MR 1313497 | Zbl 0911.18001
[6] Brown, C., Gurr, D.: A representation theorem for quantales. J. Pure Appl. Algebra 85 (1993), 1, 27–42. DOI 10.1016/0022-4049(93)90169-T | MR 1207066 | Zbl 0776.06011
[7] Chang, C. L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24 (1968), 182–190. DOI 10.1016/0022-247X(68)90057-7 | MR 0236859 | Zbl 0167.51001
[8] Dilworth, R. P.: Non-commutative residuated lattices. Trans. Amer. math. Soc. 46 (1939), 426–444. MR 0000230 | Zbl 0022.10402
[9] Gierz, G., Hofmann, K., al., et: Continuous Lattices and Domains. Cambridge University Press, 2003. MR 1975381 | Zbl 1088.06001
[10] Girard, J.: Linear logic. Theor. Comput. Sci. 50 (1987), 1–102. DOI 10.1016/0304-3975(87)90045-4 | Zbl 0647.03016
[11] Goguen, J. A.: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145–174. DOI 10.1016/0022-247X(67)90189-8 | MR 0224391 | Zbl 0145.24404
[12] Goguen, J. A.: The fuzzy Tychonoff theorem. J. Math. Anal. Appl. 43 (1973), 734–742. DOI 10.1016/0022-247X(73)90288-6 | MR 0341365 | Zbl 0278.54003
[13] Grillet, P. A.: Abstract Algebra. Second edition. Springer-Verlag, 2007. MR 2330890 | Zbl 1122.00001
[14] Gylys, R.: Involutive and relational quantaloids. Lith. Math. J. 39 (1999), 4, 376–388. DOI 10.1007/BF02465588 | MR 1803001 | Zbl 0980.18004
[15] Halmos, P.: Algebraic Logic. Chelsea Publishing Company, 1962. MR 0131961 | Zbl 0101.01101
[16] Herrlich, H., Strecker, G. E.: Category Theory. Third edition. Heldermann Verlag, 2007. MR 2377903 | Zbl 1125.18300
[17] Höhle, U.: Quantaloids as categorical basis for many valued mathematics. In: Abstracts of the 31st Linz Seminar on Fuzzy Set Theory (P. Cintula, E. P. Klement, L. N. Stout, eds.), Johannes Kepler Universität, Linz 2010, pp. 91–92.
[18] Hungerford, T.: Algebra. Springer-Verlag, 2003.
[19] Johnstone, P. T.: Stone Spaces. Cambridge University Press, 1982. MR 0698074 | Zbl 0499.54001
[20] Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Am. Math. Soc. 309 (1984), 1–71. MR 0756176 | Zbl 0541.18002
[21] Kasangian, S., Rosebrugh, R.: Decomposition of automata and enriched category theory. Cah. Topologie Géom. Différ. Catég. 27 (1986), 4, 137–143. MR 0885374 | Zbl 0625.68040
[22] Kelly, G. M.: Basic concepts of enriched category theory. Repr. Theory Appl. Categ. 10 (2005), 1–136. MR 2177301 | Zbl 1086.18001
[23] Kruml, D., Paseka, J.: Algebraic and categorical aAspects of quantales. In: Handbook of Algebra (M. Hazewinkel, ed.), 5, Elsevier, 2008, pp. 323–362. MR 2523454
[24] Lawvere, F. W.: Metric spaces, generalized logic and closed categories. Repr. Theory Appl. Categ. 1 (2002), 1–37. MR 1925933 | Zbl 1078.18501
[25] Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56 (1976), 621–633. DOI 10.1016/0022-247X(76)90029-9 | MR 0440482 | Zbl 0342.54003
[26] Lane, S. Mac: Categories for the Working Mathematician. Second edition. Springer-Verlag, 1998. MR 1712872
[27] Mitchell, B.: Rings with several objects. Adv. Math. 8 (1972), 1–161. DOI 10.1016/0001-8708(72)90002-3 | MR 0294454 | Zbl 0232.18009
[28] Mulvey, C.: & Rend. Circ. Mat. Palermo II (1986), 12, 99–104. MR 0853151
[29] Mulvey, C. J., Pelletier, J. W.: On the quantisation of points. J. Pure Appl. Algebra 159 (2001), 231–295. DOI 10.1016/S0022-4049(00)00059-1 | MR 1828940 | Zbl 0983.18007
[30] Mulvey, C. J., Pelletier, J. W.: On the quantisation of spaces. J. Pure Appl. Algebra 175 (2002), 1-3, 289–325. MR 1935983 | Zbl 1026.06018
[31] Paseka, J.: Quantale Modules. Habilitation Thesis, Department of Mathematics, Faculty of Science, Masaryk University, Brno 1999.
[32] Paseka, J.: A note on nuclei of quantale modules. Cah. Topologie Géom. Différ. Catégoriques 43 (2002), 1, 19–34. MR 1892106 | Zbl 1015.06017
[33] Pitts, A. M.: Applications of sup-lattice enriched category theory to sheaf theory. Proc. Lond. Math. Soc. III. 57 (1988), 3, 433–480. MR 0960096 | Zbl 0619.18005
[34] Rosenthal, K. I.: Quantales and Their Applications. Addison Wesley Longman, 1990. MR 1088258 | Zbl 0703.06007
[35] Rosenthal, K. I.: Free quantaloids. J. Pure Appl. Algebra 72 (1991), 1, 67–82. DOI 10.1016/0022-4049(91)90130-T | MR 1115568 | Zbl 0729.18007
[36] Rosenthal, K. I.: Girard quantaloids. Math. Struct. Comput. Sci. 2 (1992), 1, 93–108. DOI 10.1017/S0960129500001146 | MR 1159501 | Zbl 0761.18008
[37] Rosenthal, K. I.: Quantaloidal nuclei, the syntactic congruence and tree automata. J. Pure Appl. Algebra 77 (1992), 2, 189–205. DOI 10.1016/0022-4049(92)90085-T | MR 1149021 | Zbl 0761.18009
[38] Rosenthal, K. I.: Quantaloids, enriched categories and automata theory. Appl. Categ. Struct. 3 (1995), 3, 279–301. DOI 10.1007/BF00878445 | MR 1354679 | Zbl 0833.18002
[39] Rosenthal, K. I.: The Theory of Quantaloids. Addison Wesley Longman, 1996. MR 1427263 | Zbl 0845.18003
[40] Solovjovs, S.: Powerset operator foundations for categorically-algebraic fuzzy sets theories. In: Abstracts of the 31st Linz Seminar on Fuzzy Set Theory (P. Cintula, E. P. Klement, L. N. Stout, ed.), Johannes Kepler Universität, Linz 2010, pp. 143–151.
[41] Solovyov, S.: Completion of partially ordered sets. Discuss. Math., Gen. Algebra Appl. 27 (2007), 59–67. DOI 10.7151/dmgaa.1119 | MR 2319333 | Zbl 1141.18005
[42] Solovyov, S.: On coproducts of quantale algebras. Math. Stud. (Tartu) 3 (2008), 115–126. MR 2497770 | Zbl 1160.06007
[43] Solovyov, S.: On the category $Q$-Mod. Algebra Univers. 58 (2008), 35–58. MR 2375280 | Zbl 1145.06008
[44] Solovyov, S.: A representation theorem for quantale algebras. Contr. Gen. Alg. 18 (2008), 189–198. MR 2407586 | Zbl 1147.06010
[45] Solovyov, S.: Sobriety and spatiality in varieties of algebras. Fuzzy Sets Syst. 159 (2008), 19, 2567–2585. MR 2450327 | Zbl 1177.54004
[46] Solovyov, S.: From quantale algebroids to topological spaces: fixed- and variable-basis approaches. Fuzzy Sets Syst. 161 (2010), 9, 1270–1287. MR 2603069 | Zbl 1193.54010
[47] Solovyov, S.: On monadic quantale algebras: basic properties and representation theorems. Discuss. Math., Gen. Algebra Appl. 30 (2010), 1, 91–118. DOI 10.7151/dmgaa.1164 | MR 2762580 | Zbl 1220.03052
[48] Street, R.: Elementary cosmoi I. Lect. Notes Math. 420 (1974), 134–180. DOI 10.1007/BFb0063103 | MR 0354813 | Zbl 0325.18005
[49] Street, R.: Cauchy characterization of enriched categories. Repr. Theory Appl. Categ. 4 (2004), 1–16. MR 2048315 | Zbl 1099.18005
[50] Street, R.: Enriched categories and cohomology. Repr. Theory Appl. Categ. 14 (2005), 1–18. MR 2219705 | Zbl 1085.18010
[51] Stubbe, I.: Categorical structures enriched in a quantaloid: categories, distributors and functors. Theory Appl. Categ. 14 (2005), 1–45. MR 2122823 | Zbl 1079.18005
[52] Stubbe, I.: Categorical structures enriched in a quantaloid: Orders and ideals over a base quantaloid. Appl. Categ. Struct. 13 (2005), 3, 235–255. DOI 10.1007/s10485-004-7421-5 | MR 2167792 | Zbl 1093.06013
[53] Stubbe, I.: Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories. Cah. Topol. Géom. Différ. Catég. 46 (2005), 2, 99–121. MR 2153892 | Zbl 1086.18005
[54] Stubbe, I.: Categorical structures enriched in a quantaloid: tensored and cotensored categories. Theory Appl. Categ. 16 (2006), 283–306. MR 2223039 | Zbl 1119.18005
[55] Stubbe, I.: $\cal Q$-modules are $\cal Q$-suplattices. Theory Appl. Categ. 19 (2007), 4, 50–60. MR 2369018
[56] Ward, M.: Residuation in structures over which a multiplication is defined. Duke math. J. 3 (1937), 627–636. DOI 10.1215/S0012-7094-37-00351-X | MR 1546017 | Zbl 0018.19903
[57] Ward, M.: Structure residuation. Ann. Math. 39 (1938), 558–568. DOI 10.2307/1968634 | MR 1503424 | Zbl 0019.28902
[58] Ward, M., Dilworth, R. P.: Residuated lattices. Trans. Am. Math. Soc. 45 (1939), 335–354. DOI 10.1090/S0002-9947-1939-1501995-3 | MR 1501995 | Zbl 0021.10801
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