POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES

[1] S. Abramsky, Interaction categories (extended abstract), Proceedings of the First Imperial
College Department of Computing Workshop on Theory and Formal Methods, Springer-
Verlag, (1993), 57-69.
[2] S. Abramsky and S. Vickers, Quantales, observational logic and process semantics, Math.
Struct. Comput. Sci., 3 (1993), 161-227.
[3] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories: the joy of cats,
Repr. Theory Appl. Categ., 17 (2006), 1-507.
[4] D. Aerts, Foundations of quantum physics: a general realistic and operational approach, Int.
J. Theor. Phys., 38(1) (1999), 289-358.
[5] D. Aerts, E. Colebunders, A. Van Der Voorde and B. Van Steirteghem, State property systems
and closure spaces: a study of categorical equivalence, Int. J. Theor. Phys., 38(1) (1999),
359-385.
[6] D. Aerts, E. Colebunders, A. Van Der Voorde and B. Van Steirteghem, On the amnestic
modication of the category of state property systems, Appl. Categ. Struct., 10(5) (2002),
469-480.
[7] F. W. Anderson and K. R. Fuller, Rings and categories of modules, 2nd ed., Springer-Verlag,
1992.
[8] S. Z. Bai, Countably near PS-compactness in L-topological spaces, Iranian Journal of Fuzzy
Systems, 4(2) (2007), 89-94.
[9] S. Z. Bai, A new notion of fuzzy PS-compactness, Iranian Journal of Fuzzy Systems, 5(2)
(2008), 79-86.
[10] M. Barr, Autonomous categories. with an appendix by Po-Hsiang chu, Springer-Verlag, 1979.
[11] R. Betti and S. Kasangian, Tree automata and enriched category theory, Rend. Ist. Mat.
Univ. Trieste, 17 (1985), 71-78.
[12] G. Birkho, On the structure of abstract algebras, Proc. Cambridge Phil. Soc., 31 (1935),
433-454.
[13] G. Birkho, Rings of sets, Duke Math. J., 3 (1937), 443-454.
[14] G. Birkho, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications,
American Mathematical Society, XXV (1979).

[15] F. Borceux and G. Van Den Bossche, An essay on noncommutative topology, Topology Appl.,
31(3) (1989), 203-223.
[16] S. Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics,
Springer-Verlag, 78 (1981).
[17] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190.
[18] R. Cignoli, Quantiers on distributive lattices, Discrete Math., 96(3) (1991), 183-197.
[19] J. Crulis, Quantiers on multiplicative semilattices, Contr. Gen. Alg., 18 (2008), 31-46.
[20] D. M. Clark and B. A. Davey, Natural dualities for the working algebraist, Cambridge Studies
in Advanced Mathematics, Cambridge University Press, 57 (1998).
[21] P. M. Cohn, Universal algebra, D. Reidel Publ. Comp., 1981.
[22] M. E. Coniglio and F. Miraglia, Non-commutative topology and quantales, Stud. Log., 65(2)
(2000), 223-236.
[23] C. De Mitri and C. Guido, Some remarks on fuzzy powerset operators, Fuzzy Sets and
Systems, 126(2) (2002), 241-251.
[24] M. Demirci, Pointed semi-quantales and generalized lattice-valued quasi topological spaces,
Abstracts of the 29th Linz Seminar on Fuzzy Set Theory (E. P. Klement, S. E. Rodabaugh
and L. N. Stout, eds.), Johannes Kepler Universitat, Linz, (2008), 23-24.
[25] M. Demirci, Pointed semi-quantales and lattice-valued topological spaces, Fuzzy Sets and
Systems, 161(9) (2010), 1224-1241.
[26] J. T. Denniston, A. Melton and S. E. Rodabaugh, Lattice-valued topological systems, Abstracts
of the 30th Linz Seminar on Fuzzy Set Theory (U. Bodenhofer, B. De Baets, E. P.
Klement and S. Saminger-Platz, eds.), Johannes Kepler Universitat, Linz, (2009), 24-31.
[27] J. T. Denniston and S. E. Rodabaugh, Functorial relationships between lattice-valued topology
and topological systems, Quaest. Math., 32(2) (2009), 139-186.
[28] P. Eklund, Category theoretic properties of fuzzy topological spaces, Fuzzy Sets and Systems,
13 (1984), 303-310.
[29] P. Eklund, A comparison of lattice-theoretic approaches to fuzzy topology, Fuzzy Sets and
Systems, 19 (1986), 81-87.
[30] P. Eklund, Categorical fuzzy topology, Ph.D. thesis, Abo Akademi, 1986.
[31] M. Erne, General Stone duality, Topology Appl. 137(1-3) (2004), 125-158.
[32] A. Frascella and C. Guido, Structured lattices and ground categories of L-sets, Int. J. Math.
Math. Sci., 2005(17) (2005), 2783-2803.
[33] A. Frascella and C. Guido, Transporting many-valued sets along many-valued relations, Fuzzy
Sets and Systems, 159(1) (2008), 1-22.
[34] B. Ganter and R. Wille, Formale begrisanalyse. mathematische grundlagen, Berlin:
Springer, 1996.
[35] G. Gierz, K. H. Hofmann and etc., Continuous lattices and domains, Cambridge University
Press, 2003.
[36] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), 145-174.
[37] J. A. Goguen, The fuzzy Tychono theorem, J. Math. Anal. Appl., 43 (1973), 734-742.
[38] C. Guido, Fuzzy points and attachment, Fuzzy Sets and Systems, 161(16) (2010), 2150-2165.
[39] C. Guido, Powerset operators based approach to fuzzy topologies on fuzzy sets, Topological
and Algebraic Structures in Fuzzy Sets. A Handbook of Recent Developments in the Mathematics
of Fuzzy Sets (S. E. Rodabaugh and E. P. Klement, eds.), Trends Log. Stud. Log.
Libr., Kluwer Academic Publishers, 20 (2003), 401-413.
[40] C. Guido, Attachment between fuzzy points and fuzzy sets, Abstracts of the 30th Linz Seminar
on Fuzzy Set Theory (U. Bodenhofer, B. De Baets, E. P. Klement and S. Saminger-Platz,
eds.), Johannes Kepler Universitat, Linz, (2009), 52-54.
[41] P. Halmos, Algebraic logic I: Monadic Boolean algebras, Compos. Math., 12 (1955), 217-249.
[42] P. Halmos, Algebraic logic, Chelsea Publishing Company, 1962.
[43] I. M. Hanafy, A. M. Abd El-Aziz and T. M. Salman, Semi -compactness in intuitionistic
fuzzy topological spaces, Iranian Journal of Fuzzy Systems, 3(2) (2006), 53-62.
[44] H. Herrlich and G. E. Strecker, Category theory, 3rd ed., Sigma Series in Pure Mathematics,
Heldermann Verlag, 1 (2007).

[45] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, Mathematics
of Fuzzy Sets: Logic, Topology and Measure Theory (U. Hohle and S. E. Rodabaugh, eds.),
The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, 3 (1999), 123-272.
[46] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78 (1980),
659-673.
[47] T. Hungerford, Algebra, Springer-Verlag, 2003.
[48] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58 (1977), 559-571.
[49] B. Hutton, Products of fuzzy topological spaces, Topology Appl., 11 (1980), 59-67.
[50] B. Hutton, Uniformities on fuzzy topological spaces II, Fuzzy Math., 3(1) (1983), 27-34.
[51] J. R. Isbell, Atomless parts of spaces, Math. Scand., 31 (1972), 5-32.
[52] J. John and T. Baiju, Metacompactness in L-topological spaces, Iranian Journal of Fuzzy
Systems, 5(3) (2008), 71-79.
[53] P. T. Johnstone, Stone spaces, Cambridge University Press, 1982.
[54] A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Am.
Math. Soc., 309 (1984), 1-71.
[55] G. M. Kelly, Basic concepts of enriched category theory, Cambridge University Press, 1982.
[56] J. C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc., 13(III) (1963), 71-89.
[57] W. Kotze, Uniform spaces, Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory
(U. Hohle and S. E. Rodabaugh, eds.), The Handbooks of Fuzzy Sets Series, Kluwer Academic
Publishers(English), 3 (1999), 553-580 .
[58] S. Krajci, A categorical view at generalized concept lattices, Kybernetika, 43(2) (2007), 255-
264.
[59] D. Kruml, Spatial quantales, Appl. Categ. Struct., 10(1) (2002), 49-62.
[60] D. Kruml and J. Paseka, Algebraic and categorical aspects of quantales, Handbook of Algebra
(M. Hazewinkel, eds.), Elsevier, 5 (2008), 323-362.
[61] T. Kubiak, On fuzzy topologies, Ph.D. thesis, Adam Mickiewicz University, Poznan, Poland,
1985.
[62] T. Kubiak and A. Sostak, Foundations of the theory of (L;M)-fuzzy topological spaces, Abstracts
of the 30th Linz Seminar on Fuzzy Set Theory (U. Bodenhoer, B. De Baets, E. P.
Klement and S. Saminger-Platz, eds.), Johannes Kepler Universitat, Linz, (2009), 70-73.
[63] S. P. Li, Z. Fang and J. Zhao, P2-connectedness in L-topological spaces, Iranian Journal of
Fuzzy Systems, 2(1) (2005), 29-36.
[64] F. E. J. Linton, Some aspects of equational categories, Proc. Conf. Categor. Algebra, La
Jolla, (1965), 84-94.
[65] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56 (1976),
621-633.
[66] S. Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, 1998.
[67] E. G. Manes, Algebraic theories, Springer-Verlag, 1976.
[68] C. J. Mulvey, &, Rend. Circ. Mat. Palermo, II(12) (1986), 99-104.
[69] C. J. Mulvey and J. W. Pelletier, On the quantisation of points, J. Pure Appl. Algebra, 159
(2001), 231-295.
[70] C. J. Mulvey and J. W. Pelletier, On the quantisation of spaces, J. Pure Appl. Algebra,
175(1-3) (2002), 289-325.
[71] D. Papert and S. Papert, Sur les treillis des ouverts et les paratopologies, Semin. de Topologie
et de Geometrie dierentielle Ch. Ehresmann 1 (1957/58), 1 (1959), 1-9.
[72] J. Paseka, Quantale modules, Habilitation Thesis, Department of Mathematics, Faculty of
Science, Masaryk University Brno, 1999.
[73] J. Paseka, A note on nuclei of quantale modules, Cah. Topologie Geom. Dier. Categoriques,
43(1) (2002), 19-34.
[74] J. Picado, A. Pultr and A. Tozzi, Locales, Categorical Foundations: Special Topics in Order,
Topology, Algebra, and Sheaf Theory (M. C. Pedicchio and W. Tholen, eds.), Cambridge
University Press, (2004), 49-101.
[75] A. M. Pitts, Applications of sup-lattice enriched category theory to sheaf theory, Proc. Lond.
Math. Soc., III. Ser., 57(3) (1988), 433-480.
[76] V. Pratt, Chu spaces, School on category theory and applications. Lecture notes of courses,
Coimbra, Portugal, July 13-17, 1999. Coimbra: Universidade de Coimbra, Departamento de
Matematica. Textos Mat., Ser. B. 21, (1999), 39-100.
[77] G. Preu, Convenient topology, Math. Jap., 47(1) (1998), 171-183.
[78] H. A. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull.
Lond. Math. Soc., 2 (1970), 186-190.
[79] P. M. Pu and Y. M. Liu, Fuzzy topology II: Product and quotient spaces, J. Math. Anal.
Appl., 77 (1980), 20-37.
[80] A. Pultr, Frames, Handbook of Algebra (M. Hazewinkel, eds.), North-Holland Elsevier, 3
(2003), 789-858.
[81] P. Resende, Observational system specication, Selected Papers of ISCORE'94 (R. Wieringa
and R. Feenstra, eds.), World Scientic, Singapore, (1995), 135-151.
[82] P. Resende, Quantales, nite observations and strong bisimulation, Theor. Comput. Sci.,
254(1-2) (2001), 95-149.
[83] S. E. Rodabaugh, A categorical accommodation of various notions of fuzzy topology, Fuzzy
Sets and Systems, 9 (1983), 241-265.
[84] S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems, 40(2) (1991),
297-345.
[85] S. E. Rodabaugh, Categorical frameworks for stone representation theories, Applications of
Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement and U. Hohle, eds.), Theory
and Decision Library: Series B: Mathematical and Statistical Methods, Kluwer Academic
Publishers, 14 (1992), 177-231.
[86] S. E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic (poslat)
fuzzy set theories and topologies, Quaest. Math., 20(3) (1997), 463-530.
[87] S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, Mathematics of
Fuzzy Sets: Logic, Topology and Measure Theory (U. Hohle and S. E. Rodabaugh, eds.), The
Handbooks of Fuzzy Sets Series, Dordrecht: Kluwer Academic Publishers, 3 (1999), 273-388.
[88] S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies,
Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. Hohle and S. E. Rodabaugh,
eds.), The Handbooks of Fuzzy Sets Series, Dordrecht: Kluwer Academic Publishers,
3 (1999), 91-116.
[89] S. E. Rodabaugh, Axiomatic foundations for uniform operator quasi-uniformities, Topological
and Algebraic Structures in Fuzzy Sets. A Handbook of Recent Developments in the
Mathematics of Fuzzy Sets (S. E. Rodabaugh and E. P. Klement, eds.), Trends Log. Stud.
Log. Libr., Dordrecht: Kluwer Academic Publishers, 20 (2003), 199-233.
[90] S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topological
theories for lattice-valued mathematics, Int. J. Math. Math. Sci., 2007 (2007), 1-71.
[91] S. E. Rodabaugh, Functorial comparisons of bitopology with topology and the case for redun-
dancy of bitopology in lattice-valued mathematics, Appl. Gen. Topol., 9(1) (2008), 77-108.
[92] K. I. Rosenthal, Quantales and their applications, Pitman Research Notes in Mathematics,
Addison Wesley Longman, 234 (1990).
[93] K. I. Rosenthal, Free quantaloids, J. Pure Appl. Algebra, 72(1) (1991), 67-82.
[94] K. I. Rosenthal, The theory of quantaloids, Pitman Research Notes in Mathematics, Addison
Wesley Longman, 348 (1996).
[95] J. Rosicky, Equational categories, Cah. Topol. Geom. Dier., 22 (1981), 85-95.
[96] J. Rosicky, Quantaloids for concurrency, Appl. Categ. Struct., 9(4) (2001), 329-338.
[97] A. P. Shostak, Two decades of fuzzy topology: basic ideas, notions, and results, Russ. Math.
Surv., 44(6) (1989), 125-186.
[98] S. Solovjovs, From quantale algebroids to topological spaces, Abstracts of the 29th Linz Seminar
on Fuzzy Set Theory (E. P. Klement, S. E. Rodabaugh and L. N. Stout, eds.), Johannes
Kepler Universitat, Linz, (2008), 98-101.
[99] S. Solovjovs, On a categorical generalization of the concept of fuzzy set: basic denitions,
properties, examples, VDM Verlag Dr. Muller, 2008.

[100] S. Solovjovs, Embedding topology into algebra, Abstracts of the 30th Linz Seminar on Fuzzy
Set Theory (U. Bodenhofer, B. De Baets, E. P. Klement and S. Saminger-Platz, eds.), Johannes
Kepler Universitat, Linz, (2009), 106-110.
[101] S. Solovyov, Categorical foundations of variety-based topology and topological systems, Submitted.
[102] S. Solovyov, Categorically-algebraic frameworks for Priestley duality, Contr. Gen. Alg., 19
(2010), 187-208.
[103] S. Solovyov, Composite variety-based topological theories, Submitted.
[104] S. Solovyov, Generalized fuzzy topology versus non-commutative topology, To Appear in
Fuzzy Sets and Systems, doi: 10. 1016/ j. fss. 2011. 03. 005.
[105] S. Solovyov, Localication of variable-basis topological systems, To Appear in Quaest. Math.
[106] S. Solovyov, On a generalization of the concept of state property system, To Appear in Soft
Comput.
[107] S. Solovyov, On algebraic and coalgebraic categories of variety-based topological systems, To
Appear in Iranian Journal of Fuzzy Systems.
[108] S. Solovyov, On monadic quantale algebras: basic properties and representation theorems,
Discuss. Math., Gen. Algebra Appl., 30(1) (2010), 91-118.
[109] S. Solovyov, Variable-basis topological systems versus variable-basis topological spaces, Soft
Comput., 14(10) (2010), 1059-1068.
[110] S. Solovyov, Categorical frameworks for variable-basis sobriety and spatiality, Math. Stud.
(Tartu), 4 (2008), 89-103.
[111] S. Solovyov, On coproducts of quantale algebras, Math. Stud. (Tartu), 3 (2008), 115-126.
[112] S. Solovyov, On the category Q-Mod, Algebra Univers., 58 (2008), 35-58.
[113] S. Solovyov, A representation theorem for quantale algebras, Contr. Gen. Alg., 18 (2008),
189-198.
[114] S. Solovyov, Sobriety and spatiality in varieties of algebras, Fuzzy Sets and Systems,
159(19) (2008), 2567-2585.
[115] S. Solovyov, From quantale algebroids to topological spaces: xed- and variable-basis ap-
proaches, Fuzzy Sets and Systems, 161 (2010), 1270-1287.
[116] M. H. Stone, The theory of representations for Boolean algebras, Trans. Am. Math. Soc.,
40 (1936), 37-111.
[117] M. H. Stone, Topological representations of distributive lattices and Brouwerian logics, Cas.
Mat. Fys., 67 (1937), 1-25.
[118] S. Vickers, Topology via logic, Cambridge University Press, 1989.
[119] G. F. Wen, F. G. Shi and H. Y. Li, Almost S-compactness in L-topological spaces, Iranian
Journal of Fuzzy Systems, 5(3) (2008), 31-44.
[120] S. Willard, General topology. reprint of the 1970 original, Dover Publications, 2004.
[121] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6(1) (2009),
63-80.
[122] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-365.