LATTICE-VALUED CATEGORIES OF LATTICE-VALUED CONVERGENCE SPACES

Document Type: Research Paper

Author

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

Abstract

We study L-categories of lattice-valued convergence spaces. Such
categories are obtained by \fuzzifying" the axioms of a lattice-valued convergence
space. We give a natural example, study initial constructions and
function spaces. Further we look into some L-subcategories. Finally we use
this approach to quantify how close certain lattice-valued convergence spaces
are to being lattice-valued topological 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 J. Al-Mufarrij, Frame valued strati ed generalized convergence
groups, Quaest. Math., 31 (2008), 279-302.
[3] A. Craig and G. Jager, A common framework for lattice-valued uniform spaces and proba-
bilistic uniform convergence spaces, Fuzzy Sets and Systems, 160 (2009), 1177-1203.
[4] U. Hohle, Commutative, residuated L-monoids, In: Non-classical Logics and Their Applications
to Fuzzy Subsets (U. Hohle, E.P. Klement, eds.), Kluwer, Dordrecht, (1995), 53-106.
[5] U. Hohle, Locales and L-topologies, In: Categorical Methods in Algebra and Topology (H. E.
Porst, eds.), Mathematik-Arbeitspapiere 48, Univ. Bremen, (1997), 223-250.
[6] 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,
eds.), Kluwer, Dordrecht, (1999), 123-272.
[7] G. Jager, Fuzzy properties in fuzzy convergence spaces, Int. J. Math. Math. Sci., 29 (2002),
737-748.
[8] G. Jager, A category of L-fuzzy convergence spaces, Quaest. Math., 24 (2001), 501-517.
[9] G. Jager, Subcategories of lattice valued convergence spaces, Fuzzy Sets and Systems, 156
(2005), 1-24.
[10] G. Jager, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and
Systems, 158 (2007), 424-435.
[11] G. Jager, Fischer's diagonal condition for lattice-valued convergence spaces, Quaest. Math.,
31 (2008), 11-25.

[12] G. Jager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159
(2008), 2488-2502.
[13] G. Jager, Compacti cation of lattice-valued convergence spaces, Fuzzy Sets and Systems, 161
(2010), 1002-1010.
[14] T. Kubiak and A. P. Sostak , A fuzzi cation of the category of M-valued L-topological spaces,
App. Gen. Top., 5 (2004), 137-154.
[15] E. Lowen and R. Lowen, On measures of compactness in fuzzy topological spaces, J. Math.
Anal. Appl., 131 (1988), 329-340.
[16] H. Poppe, Compactness in general function spaces, VEB Deutscher Verlag Der Wissenschaften,
Berlin, 1974.
[17] A. Sostak, On compactness and connectedness degrees of fuzzy sets in fuzzy topological spaces,
In: General Topology and Its Relations to Modern Analysis and Algebra, VI (Prague, 1986),
Res. Exp. Math., Heldermann, Berlin, 16 (1988), 519-532.
[18] A. Sostak, Fuzzy categories versus categories of fuzzily structured sets: elements of the theory
of fuzzy categories, In: Categorical Methods in Algebra and Topology (H. E. Porst, eds.),
Mathematik-Arbeitspapiere 48, Univ. Bremen, (1997), 407-437.
[19] A. Sostak, Fuzzy categories related to algebra and topology, Tatra Mt. Math. Publ., 16 (1999),
159-185.
[20] A. P. Sostak, Fuzzy functions and an extension of the category L-TOP of Chang-Goguen L-
topological spaces, In: Proceedings of the 9th Prague Topological Symposium (Prague 2001),
Topology ATLAS, Totoronto, (2002), 271-294.
[21] A. P. Sostak, On some fuzzy categories related to category L-TOP of L-topological spaces,
In: Topological and Algebraic Structures in Fuzzy Sets (eds., S. E. Rodabaugh and E. P.
Klement), Kluwer, (2003), 337-372.
[22] A. P. Sostak, L-valued categories: generalities and examples related to algebra and topology,
In: Categorical Structures and Their Applications (eds., W. Gahler and G. Preuss), World
Scienti c, (2004), 291-312.
[23] W. Yao, On many-valued strati ed 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.
[25] M. S. Ying, A new approach to fuzzy topology, part I, Fuzzy Sets and Systems, 39 (1991),
303-321.
[26] Y. Yue and J. Fang, Fuzzy ideals and fuzzy limit structures, Iranian Journal of Fuzzy Systems,
5 (2008), 55-64.