International Journal of Mathematics and Mathematical Sciences

Volume 2007 (2007), Article ID 43645, 71 pages

http://dx.doi.org/10.1155/2007/43645

Research Article

## Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

Department of Mathematics and Statistics, Youngstown State University, Youngstown, OH 44555-3609, USA

Received 15 September 2004; Revised 8 March 2006; Accepted 2 April 2006

Academic Editor: Robert Lowen

Copyright © 2007 S. E. Rodabaugh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- U. Höhle and A. P. Šostak, “Axiomatic foundations of fixed-basis fuzzy topology,” in
*Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory*, U. Höhle and S. E. Rodabaugh, Eds., vol. 3 of*The Handbooks of Fuzzy Sets Series*, pp. 123–272, Kluwer Academic Publishers, Boston, Mass, USA, 1999, chapter 3. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Categorical foundations of variable-basis fuzzy topology,” in
*Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory*, U. Höhle and S. E. Rodabaugh, Eds., vol. 3 of*The Handbooks of Fuzzy Sets Series*, pp. 273–388, Kluwer Academic Publishers, Boston, Mass, USA, 1999, chapter 4. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. G. Manes,
*Algebraic Theories*, Springer, New York, NY, USA, 1976. View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Necessary and sufficient conditions for powersets in $\mathbf{S}\mathbf{e}\mathbf{t}$ and $\mathbf{S}\mathbf{e}\mathbf{t}\times \mathbf{C}$ to form algebraic theories,” in
*Fuzzy Logics and Related Structures: Abstracts of 26th Linz Seminar on Fuzzy Set Theory*, S. Gottwald, P. Hájek, U. Höhle, and E. P. Klement, Eds., pp. 89–97, Universitätsdirektion Johannes Kepler Universtät, Bildungszentrum St. Magdalena, Linz, Austria, February 2005, (A-4040, Linz). View at Google Scholar - S. E. Rodabaugh, “Relationship of algebraic theories to powersets over objects in $\mathbf{S}\mathbf{e}\mathbf{t}$ and $\mathbf{S}\mathbf{e}\mathbf{t}\times \mathbf{C}$,” preprint.
- J. Adámek, H. Herrlich, and G. E. Strecker,
*Abstract and Concrete Categories: The Joy of Cats*, Pure and Applied Mathematics (New York), John Wiley & Sons, New York, NY, USA, 1990. View at Zentralblatt MATH · View at MathSciNet - C. De Mitri and C. Guido, “$G$-fuzzy topological spaces and subspaces,”
*Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento*, no. 29, pp. 363–383, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. De Mitri and C. Guido, “Some remarks on fuzzy powerset operators,”
*Fuzzy Sets and Systems*, vol. 126, no. 2, pp. 241–251, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. De Mitri, C. Guido, and R. E. Toma, “Fuzzy topological properties and hereditariness,”
*Fuzzy Sets and Systems*, vol. 138, no. 1, pp. 127–147, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Frascella and C. Guido, “Structural lattices and ground categories of $L$-sets,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 17, pp. 2783–2803, 2005, in \cite{28}, 53–54. View at Publisher · View at Google Scholar · View at MathSciNet - A. Frascella and C. Guido, “Topological categories of $L$-sets and $\left(L,M\right)$-topological spaces on structured lattices,” in
*Fuzzy Logics and Related Structures: Abstracts of 26th Linz Seminar on Fuzzy Set Theory*, S. Gottwald, P. Hájek, U. Höhle, and E. P. Klement, Eds., p. 50, Universitätsdirektion Johannes Kepler Universtät, Bildungszentrum St. Magdalena, Linz, Austria, February 2005, (A-4040, Linz). View at Google Scholar - C. Guido, “The subspace problem in the traditional point set context of fuzzy topology,”
*Quaestiones Mathematicae*, vol. 20, no. 3, pp. 351–372, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Guido, “Powerset operators based approach to fuzzy topologies on fuzzy sets,” in
*Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets*, vol. 20 of*Trends in Logic—Studia Logica Library*, pp. 401–413, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, chapter 15. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Höhle,
*Many Valued Topology and Its Applications*, Kluwer Academic Publishers, Boston, Mass, USA, 2001. View at Zentralblatt MATH · View at MathSciNet - G. Birkhoff,
*Lattice Theory*, vol. 25 of*American Mathematical Society Colloquium Publications*, American Mathematical Society, Providence, RI, USA, 3rd edition, 1967. View at Zentralblatt MATH · View at MathSciNet - C. J. Mulvey and M. Nawaz, “Quantales: quantal sets,” in
*Non-Classical Logics and Their Applications to Fuzzy Subsets (Linz, 1992)*, U. Höhle and E. P. Klement, Eds., vol. 32 of*Theory and Decision Library. Series B: Mathematical and Statistical Methods*, pp. 159–217, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. I. Rosenthal,
*Quantales and Their Applications*, vol. 234 of*Pitman Research Notes in Mathematics*, Longman Scientific & Technical, Burnt Mill, Harlow, UK, 1990. View at Zentralblatt MATH · View at MathSciNet - P. T. Johnstone,
*Stone Spaces*, vol. 3 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1982. View at Zentralblatt MATH · View at MathSciNet - E. P. Klement, R. Mesiar, and E. Pap,
*Triangular Norms*, vol. 8 of*Trends in Logic—Studia Logica Library*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Powerset operator based foundation for point-set lattice-theoretic (poslat) fuzzy set theories and topologies,”
*Quaestiones Mathematicae*, vol. 20, no. 3, pp. 463–530, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Powerset operator foundations for poslat fuzzy set theories and topologies,” in
*Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory*, U. Höhle and S. E. Rodabaugh, Eds., vol. 3 of*The Handbooks of Fuzzy Sets Series*, pp. 91–116, Kluwer Academic Publishers, Boston, Mass, USA, 1999, chapter 2. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Axiomatic foundations for uniform operator quasi-uniformities,” in
*Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets*, vol. 20 of*Trends in Logic—Studia Logica Library*, pp. 199–233, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, chapter 7. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “A categorical accommodation of various notions of fuzzy topology: preliminary report,” in
*Proceedings of the 3rd International Seminar on Fuzzy Set Theory*, E. P. Klement, Ed., vol. 3, pp. 119–152, Universitätsdirektion Johannes Kepler Universtät, Linz, Austria, 1981, (A-4040). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “A categorical accommodation of various notions of fuzzy topology,”
*Fuzzy Sets and Systems*, vol. 9, no. 3, pp. 241–265, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Point-set lattice-theoretic topology,”
*Fuzzy Sets and Systems*, vol. 40, no. 2, pp. 297–345, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Categorical frameworks for Stone representation theories,” in
*Applications of Category Theory to Fuzzy Subsets*, S. E. Rodabaugh, E. P. Klement, and U. Höhle, Eds., vol. 14 of*Theory and Decision Library. Series B: Mathematical and Statistical Methods*, pp. 177–231, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992, chapter 7. View at Google Scholar · View at MathSciNet - S. E. Rodabaugh, “A point-set lattice-theoretic framework $\mathbb{T}$ which contains $LOC$ as a subcategory of singleton spaces and in which there are general classes of Stone representation and compactification theorems,” February 1986 / April 1987, Youngstown State University Central Printing Office, Youngstown, Ohio, USA.
- S. Jenei, “Structure of Girard monoids on $\left[0,1\right]$,” in
*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., vol. 20 of*Trends in Logic—Studia Logica Library*, pp. 277–308, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, chapter 10. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. P. Šostak, “On a fuzzy syntopogeneous structure,”
*Quaestiones Mathematicae*, vol. 20, no. 3, pp. 431–463, 1997. View at Google Scholar - P. Eklund, “Category theoretic properties of fuzzy topological spaces,”
*Fuzzy Sets and Systems*, vol. 13, no. 3, pp. 303–310, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Eklund, “A comparison of lattice-theoretic approaches to fuzzy topology,”
*Fuzzy Sets and Systems*, vol. 19, no. 1, pp. 81–87, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Eklund,
*Categorical fuzzy topology*, Doctoral Dissertation. - J. A. Goguen, “$L$-fuzzy sets,”
*Journal of Mathematical Analysis and Applications*, vol. 18, pp. 145–174, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, “Separation axioms: representation theorems, compactness, and compactifications,” in
*Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory*, vol. 3 of*The Handbooks of Fuzzy Sets Series*, pp. 481–552, Kluwer Academic Publishers, Boston, Mass, USA, 1999, chapter 7. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. L. Chang, “Fuzzy topological spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 24, no. 1, pp. 182–190, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Goguen, “The fuzzy Tychonoff theorem,”
*Journal of Mathematical Analysis and Applications*, vol. 43, pp. 734–742, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Höhle, “Upper semicontinuous fuzzy sets and applications,”
*Journal of Mathematical Analysis and Applications*, vol. 78, no. 2, pp. 659–673, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Höhle and A. P. Šostak, “A general theory of fuzzy topological spaces,”
*Fuzzy Sets and Systems*, vol. 73, no. 1, pp. 131–149, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Kotzé, “Lifting of sobriety concepts with particular reference to $\left(L,M\right)$-topological spaces,” in
*Trends in Logic—Studia Logica Library*, pp. 415–426, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, chapter 16. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kubiak,
*On fuzzy topologies*, Doctoral Dissertation. - T. Kubiak and A. P. Šostak, “Lower set-valued fuzzy topologies,”
*Quaestiones Mathematicae*, vol. 20, no. 3, pp. 423–429, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Solovyov, “On the category $\mathbf{S}\mathbf{e}\mathbf{t}\left(\mathbf{J}\mathbf{C}\mathbf{P}\mathbf{o}\mathbf{s}\right)$,”
*Fuzzy Sets and Systems*, vol. 157, no. 3, pp. 459–465, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. P. Šostak, “On a fuzzy topological structure,”
*Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento*, no. 11, pp. 89–103, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Solovyov, “Categories of lattice-valued sets as categories of arrows,”
*Fuzzy Sets and Systems*, vol. 157, no. 6, pp. 843–854, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. P. Šostak, “On a category for fuzzy topology,”
*Zbornik Radova Filozofskog Fakulteta u Nišu. Serija Matematika*, no. 2, pp. 61–67, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. P. Šostak, “On compactness and connectedness degrees of fuzzy sets in fuzzy topological spaces,” in
*General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the V-th Prague Topological Symposium (Prague, 1986)*, vol. 16 of*Res. Exp. Math.*, pp. 519–532, Heldermann, Berlin, Germany, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. P. Šostak, “Two decades of fuzzy topology: basic ideas, notions, and results,”
*Russian Mathematical Surveys*, vol. 44, no. 6, pp. 125–186, 1989. View at Google Scholar · View at Zentralblatt MATH - A. P. Šostak, “On some modifications of fuzzy topology,”
*Matematički Vesnik*, vol. 41, no. 1, pp. 51–64, 1989. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. P. Šostak, “On the neighborhood structure of fuzzy topological spaces,”
*Zbornik Radova Filozofskog Fakulteta u Nišu. Serija Matematika*, no. 4, pp. 7–14, 1990. View at Google Scholar · View at MathSciNet - A. P. Šostak, “Towards the concept of a fuzzy category,” in
*Mathematics*, vol. 562 of*Latv. Univ. Zināt. Raksti*, pp. 85–94, Latv. Univ., Riga, Latvia, 1991. View at Google Scholar · View at MathSciNet - L. A. Zadeh, “Fuzzy sets,”
*Information and Control*, vol. 8, no. 3, pp. 338–353, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. P. Šostak, “On some fuzzy categories related to category $L$-$\mathbf{T}\mathbf{O}\mathbf{P}$ of $L$-topological spaces,” in
*Trends in Logic—Studia Logica Library*, pp. 337–372, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, chapter 12. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Höhle and S. E. Rodabaugh, Eds.,
*Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory*, vol. 3 of*The Handbooks of Fuzzy Sets Series*, Kluwer Academic Publishers, Boston, Mass, USA, 1999. View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh and E. P. Klement, Eds.,
*Trends in Logic—Studia Logica Library*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at Zentralblatt MATH · View at MathSciNet - S. E. Rodabaugh, E. P. Klement, and U. Höhle, Eds.,
*Applications of Category Theory to Fuzzy Subsets*, vol. 14 of*Theory and Decision Library. Series B: Mathematical and Statistical Methods*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at Zentralblatt MATH · View at MathSciNet - U. Höhle, “Sheaves on quantales,” in
*Fuzzy Logics and Related Structures: Abstracts of 26th Linz Seminar on Fuzzy Set Theory*, S. Gottwald, P. Hájek, U. Höhle, and E. P. Klement, Eds., p. 59, Universitätsdirektion Johannes Kepler Universtät, Bildungszentrum St. Magdalena, Linz, Austria, February 2005, (A-4040, Linz). View at Google Scholar - U. Höhle, “Fuzzy sets and sheaves,” in
*Mathematics of Fuzzy Systems: Abstracts of 25th Linz Seminar on Fuzzy Set Theory*, E. P. Klement and E. Pap, Eds., pp. 68–69, Universitätsdirektion Johannes Kepler Universtät, Bildungszentrum St. Magdalena, Linz, Austria, February 2004, (A-4040, Linz), in submission. View at Google Scholar - E. P. Klement and E. Pap, Eds., “Mathematics of Fuzzy Systems: Abstracts of 25th Linz Seminar on Fuzzy Set Theory,” Universitätsdirektion Johannes Kepler Universtät, Bildungszentrum St. Magdalena, Linz, Austria, February 2004, (A-4040, Linz). View at Google Scholar