Remarks on completeness of lattice-valued Cauchy spaces

Document Type : Research Paper


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


We study different completeness definitions for two categories of lattice-valued Cauchy spaces and the relations between these definitions. We also show the equivalence of a so-called completion axiom and the existence of a completion.


[1] H. Boustique and G. Richardson, Regularity: Lattice-valued Cauchy spaces, Fuzzy Sets and
Systems, 190 (2012), 94-104.
[2] P. V. Flores, R. N. Mohapatra and G. Richardson, Lattice-valued spaces: fuzzy convergence,
Fuzzy Sets and Systems, 157 (2006), 2706-2714.
[3] 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.
[4] G. Jager, A category of L-fuzzy convergence spaces, Quaest. Math., 24 (2001), 501-517.
[5] G. Jager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156
(2005), 1-24.
[6] G. Jager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159
(2008), 2488-2502.
[7] G. Jager, Lattice-valued Cauchy spaces and completion, Quaest. Math., 33 (2010), 53-74.
[8] G. Jager, Largest and smallest T2-compacti cations of lattice-valued convergence spaces,
Fuzzy Sets and Systems, 190 (2012), 32-46.
[9] G. Jager, On diagonal completion of lattice-valued diagonal Cauchy spaces, Fuzzy Sets and
Systems, to appear.
[10] H. H. Keller, Die Limes-Uniformisierbarkeit der Limesraume, Math. Ann., 176 (1968), 334-
[11] D. C. Kent and G. D. Richardson, Completions of probabilistic Cauchy spaces, Math. Japonica,
48 (1998), 399-407.
[12] H. Nusser, A generalization of probabilistic uniform spaces, Appl. Cat. Structures, 10 (2002),
[13] B. Pang, The category of strati ed L- lter spaces, Fuzzy Sets and Systems, 247 (2014),
108-126. .
[14] E. E. Reed, Completions of uniform convergence spaces, Math. Ann., 194 (1971), 83-108.
[15] X. F. Yang and S. G. Li, Completion of strati ed (L,M)- lter tower spaces, Fuzzy Sets
Systems, 210 (2013), 22-38.