A NOTE ON STRATIFIED LM-FILTERS

Document Type: Research Paper

Author

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

Abstract

We develop a theory of stratified $LM$-filters which generalizes the theory of stratified $L$-filters. Our stratification condition explicitly depends on a suitable mapping between the lattices $L$ and $M$. If $L$ and $M$ are identical and the mapping is the identity mapping, then we obtain the theory of stratified $L$-filters. Based on the stratified $LM$-filters, a general theory of lattice-valued convergence spaces can be developed.

Keywords


bibitem{adamek-herrlich-strecker1989}
J. Ad'{a}mek., H. Herrlich  and G. E. Strecker, {it Abstract and concrete categories}, Wiley, New York, 1989.

bibitem{fang2010}
J. Fang, {it Stratified L-ordered convergence structures}, Fuzzy Sets and Systems, {bf 161} (2010), 2130 -- 2149.

bibitem{flores-mohapatra-richardson2006}
P. V. Flores, R. N. Mohapatra and G. Richardson, {it Lattice-valued spaces: fuzzy convergence}, Fuzzy Sets and Systems, {bf 157} (2006), 2706 -- 2714.

bibitem{gaehler2003}
W. G"ahler, {it Monadic convergence structures}, In: S. E. Rodabaugh and E. P. Klement, eds., Topological and Algebraic Structures in Fuzzy Sets, Kluwer, (2003), 57--79.

bibitem{hoehle1997}
U. H"ohle, {it Locales and L-topologies}, In: Categorical Methods in Algebra and Topology -- a Collection of Papers in Honour of Horst Herrlich, Mathematik-Arbeitspapiere, {bf 48} (1997), 233 -- 250.

bibitem{hoehle1999}
U. H"ohle, {it Characterization of $L$-topologies by $L$-valued neighbourhoods}, In: Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory (U. H"ohle, S. E. Rodabaugh, eds.), Kluwer, Boston/Dordrecht/London, (1999), 389--432.

bibitem{hoehle-sostak1999}
U. H"ohle and A. P. Sostak, {it Axiomatic foundations of fixed-basis fuzzy topology}, In: Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory (U. H"ohle, S. E. Rodabaugh, eds., Kluwer, Boston/Dordrecht/London, (1999), 123--272.

bibitem{jaeger2001}
G. J"ager, {it A category of L-fuzzy convergence spaces}, Quaest. Math., {bf 24} (2001), 501--517.

bibitem{jaeger2005}
G. J"ager, {it Subcategories of lattice-valued convergence spaces}, Fuzzy Sets and Systems, {bf 156} (2005), 1--24.

bibitem{jaeger2007}
G. J"ager, {it Pretopological and topological lattice-valued convergence spaces}, Fuzzy Sets and Systems, {bf 158} (2007), 424--435.

bibitem{jaeger2011}
G. J"ager, {it Lattice-valued categories of lattice-valued convergence spaces}, Iranian Journal  of Fuzzy Systems, {bf 8}{bf (2)} (2011), 67--89.

bibitem{lowen-lowen1992}
E. Lowen and R. Lowen, {it A topological universe extension of FTS}, In: Applications of Category Theory to Fuzzy Sets (S. E. Rodabaugh, E. P. Klement, U. H"ohle, eds.), Kluwer, 1992.

bibitem{lowen-lowen-wuyts1991}
E. Lowen, R. Lowen and P. Wuyts, {it The categorical topology approach to fuzzy topology and fuzzy convergence}, Fuzzy Sets and Systems, {bf 40} (1991), 347--373.


bibitem{preuss1995}
G. Preuss, {it Semiuniform convergence spaces}, Math. Japonica, {bf 41} (1995), 465--491.

bibitem{richardson-kent1996}
G. D. Richardson and D. C. Kent, {it Probabilistic convergence spaces}, Journal of the Australian Math. Soc. A, {bf 61} (1996), 400--420.

bibitem{yao2008}
W. Yao, {it On many-valued $L$-fuzzy convergence spaces}, Fuzzy Sets and Systems, {bf 159} (2008), 2503 -- 2519.

bibitem{yao2009}
W. Yao, {it On $L$-fuzzifying convergence spaces}, Iranian Journal of Fuzzy Systems, {bf 6} (2009), 63--80.

bibitem{yao2012}
W. Yao, {it Moore-Smith convergence in (L,M)-fuzzy topology}, Fuzzy Sets and Systems, {bf 190} (2012), 47--62.