A COMMON FRAMEWORK FOR LATTICE-VALUED, PROBABILISTIC AND APPROACH UNIFORM (CONVERGENCE) SPACES

Document Type : Research Paper

Author

School of Mechanical Engineering, University of Applied Sciences Stralsund, 18435 Stralsund, Germany

Abstract

We develop a general framework for various lattice-valued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$-stratified $LM$-filter, where $L$ and $M$ are suitable frames. A stratified $LMN$-uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the lattice-valued, probabilistic and approach uniform convergence spaces as examples. We show that the resulting category $sLMN$-$UCTS$ is topological, well-fibred and Cartesian closed. We furthermore define stratified $LMN$-uniform tower spaces and show that the category of these spaces is isomorphic to the subcategory of stratified $LMN$-principal uniform convergence tower spaces. Finally we study the underlying stratified $LMN$-convergence tower spaces.

Keywords


[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New
York 1989.
[2] T. M. G. Ahsanullah and G. Jager, Probabilistic uniform convergence spaces rede ned, Acta
Math. Hungar., 146 (2015), 376 { 390.
[3] N. Bourbaki, General topology, Chapters 1 { 4, Springer Verlag, Berlin - Heidelberg - New
York - London - Paris - Tokyo, 1990.
[4] M. H. Burton, M. A. de Prada Vicente and J. Gutierrez Garca, Generalized uniform spaces,
J. Fuzzy Math., 4 (1996), 363 { 380.
[5] C. H. Cook and H. R. Fischer, Uniform convergence structures, Math. Ann. 173 (1967), 290
{ 306.
[6] A. Craig and G. Jager, A common framework for lattice-valued uniform spaces and proba-
bilistic uniform limit spaces, Fuzzy Sets and Systems, 160(2009), 1177 { 1203.
[7] J. Fang, Lattice-valued semiuniform convergence spaces, Fuzzy Sets and Systems, 195 (2012),
33 { 57.
[8] R. C. Flagg, Quantales and continuity spaces, Algebra Univers., 37 (1997), 257 { 276.
[9] L. C. Florescu, Probabilistic convergence structures, Aequationes Math., 38 (1989), 123 {
145.
[10] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A com-
pendium of continuous lattices, Springer-Verlag Berlin Heidelberg, 1980.
[11] J. Gutierrez Garca, A uni ed approach to the concept of fuzzy L-uniform space, Thesis,
Universidad del Pais Vasco, Bilbao, Spain, 2000.
[12] J. Gutierrez Garca, M. A. de Prada Vicente and A. P. Sostak, A uni ed approach to the
concept of fuzzy L-uniform space, In: S. E. Rodabaugh, E. P. Klement (Eds.), Topological
and algebraic structures in fuzzy sets, Kluwer, Dordrecht, (2003), 81 { 114.
[13] U. Hohle, Characterization of L-topologies by L-valued neighborhoods, In: U. Hohle, S.E.
Rodabauch (Eds.), Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory, Kluwer,
Boston/Dordrecht/London 1999, 389 { 432.
[14] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, In: U. Hohle,
S. E. Rodabauch (Eds.), Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory,
Kluwer, Boston/Dordrecht/London 1999, 123 { 272.
[15] G. Jager, A category of L-fuzzy convergence spaces, Quaestiones Math., 24 (2001), 501 { 518.
[16] G. Jager, Fischer's diagonal condition for lattice-valued convergence spaces, Quaestiones
Math., 31 (2008), 11 { 25.
[17] G. Jager, A note on strati ed LM- lters, Iranian Journal of Fuzzy Systems, 10(4) (2013),
135 { 142.
[18] G. Jager, Strati ed LMN-convergence tower spaces, Fuzzy Sets and Systems, 282 (2016), 62
{ 73.
[19] G. Jager, Uniform connectedness and uniform local connectedness for lattice-valued uniform
convergence spaces, Iranian Journal of Fuzzy Systems, 13(3) (2016), 95 { 111.
[20] G. Jager and M. H. Burton, Strati ed L-uniform convergence spaces, Quaestiones Math., 28
(2005), 11 { 36.
[21] Y. J. Lee and B. Windels, Transitivity in uniform approach theory, Int. J. Math. and Math.
Sci., 32 (2002), 707 { 720.
[22] E. Lowen, R. Lowen and P. Wuyts, The categorical topology approach to fuzzy topology and
fuzzy convergence, Fuzzy Sets and Systems, 40 (1991), 347 { 373.
[23] R. Lowen and B. Windels, AUnif: A commmon supercategory of pMET and Unif, Int. J.
Math. and Math. Sci., 21 (1998), 1 { 18.
[24] H. Nusser, A generalization of probabilistic uniform spaces, Appl. Cat. Structures, 10 (2002),
81 { 98.
[25] G. Preuss, Foundations of topology - An Approach to Convenient Topology, Kluwer, Dordrecht
2002.
[26] B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.
[27] 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), 591 { 637.