CONVERGENCE APPROACH SPACES AND APPROACH SPACES AS LATTICE-VALUED CONVERGENCE SPACES

Document Type : Research Paper

Author

Department of Statistics, Rhodes University, 6140 Grahamstown, South Africa

Abstract

We show that the category of convergence approach spaces is a
simultaneously reective and coreective subcategory of the category of latticevalued
limit spaces. Further we study the preservation of diagonal conditions,
which characterize approach spaces. It is shown that the category of preapproach
spaces is a simultaneously reective and coreective subcategory of
the category of lattice-valued pretopological spaces and that the category of
approach spaces is a coreective subcategory of a category of lattice-valued
topological convergence spaces

Keywords

References

[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New
York, 1989.
[2] P. Brock and D. C. Kent, Approach spaces, limit tower spaces, and probabilistic convergence
spaces, Applied Categorical Structures, 5 (1997), 99-110.
[3] P. V. Flores, R. N. Mohapatra and G. Richardson, Lattice-valued spaces: fuzzy convergence,
Fuzzy Sets and Systems, 157 (2006), 2706- 2714.
[4] W. Gahler, Monadic convergence structures, In S. E. Rodabaugh and E. P. Klement, Editors,
Topological and Algebraic Structures in Fuzzy Sets. Kluwer Academic Publishers, Dordrecht,
2003.
[5] J. Gutierrez Garca, I. Mardones Perez and M. H. Burton, The relationship between various
lter notions on a GL-monoid, J. Math. Anal. Appl., 230 (1999), 291-302.
[6] U. Hohle, Commutative, residuated l-monoids, In: Non-classical Logics and Their Application
to Fuzzy Subsets (U. Hohle, S. E. Rodabaugh and eds.), Kluwer, Dordrecht, (1995), 53-106.
[7] U. Hohle, Many valued topology and its applications, Kluwer, Boston/Dordrecht/London,
2001.
[8] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, In: Mathematics
of Fuzzy Sets. Logic, Topology and Measure Theory (U. Hohle, S. E. Rodabaugh and
eds.), Kluwer, Boston/Dordrecht/London, (1999), 123-272.
[9] G. Jager, A category of L-fuzzy convergence spaces, Quaest. Math., 24 (2001), 501-517.
[10] G. Jager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156
(2005), 1-24.
[11] G. Jager, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and
Systems, 158 (2007), 424-435.
[12] G. Jager, Fischer's diagonal condition for lattice-valued convergence spaces, Quaest. Math.,
31 (2008), 11-25.
[13] G. Jager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159
(2008), 2488-2502.
[14] G. Jager, Lattice-valued categories of lattice-valued convergence spaces, Iranian Journal of
Fuzzy Systems, 8 (2011), 67-89.
[15] E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Dordrecht, 2000.
[16] H. J. Kowalsky, Limesraume und Komplettierung, Math. Nachrichten, 12 (1954), 301-340.
[17] L. Li and Q. Jin, On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets and Systems,
doi:10.1016/j.fss.2010.10.002, to appear.
[18] E. Lowen and R. Lowen, A quasitopos containing CONV and MET as full subcategories,
Internat. J. Math. and Math. Sci., 11 (1988), 417- 438.
[19] R. Lowen, Approach spaces: a common supercategory of TOP and MET, Math. Nachr., 141
(1989), 183-226.
[20] R. Lowen, Approach spaces: the missing link in the topology-uniformity-metric triad, Claredon
Press, Oxford, 1997.
[21] D. L. Orpen and G. Jager, Lattice-valued convergence spaces: extending the lattice context,
Fuzzy Sets and Systems, 190 (2012), 1-20.
[22] B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.
[23] W. Yao, On many-valued L-fuzzy convergence spaces, Fuzzy Sets and Systems, 159 (2008),
2503-2519.
[24] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6 (2009),
63-80.
Iranian Journal of Fuzzy Systems
Volume 9, Issue 4 - Serial Number 4
September and October 2012
Pages 1-16

Files

Share

How to cite

Statistics