Document Type : Research Paper


Department of Mathematics, Ocean University of China, 238 Songling Road, 266100, Qingdao, P.R.China


This note studies the relationship between Hutton's quasi-uniformities and Shi's quasi-uniformities. It is shown that when $L$ satisfies``multiple choice principle" for co-prime elements, the category of Hutton's quasi-uniform spaces is a bireflective full subcategory of the category of Shi's quasi-uniform spaces. Especially, if the remote-neighborhood mapping defined by Shi preserves arbitrary joins, then the two categories are isomorphic to each other.


\bibitem{Adamek} J. Ad\'{a}mek, H. Herrlich and G. E. Strecker, {\it Abstract and concrete categories}, J. Wiley \& Sons, New York, 1990.

\bibitem{Fle} P. Fletcher and W. F. Lindgren, {\it Quasi-uniform spaces}, Marcel Dekker, New York, 1982.

\bibitem{Gie} G. Gierz and et.al, {\it Continuous lattices and domains}, Encyclopedia of Mathematics and Its Applications,
Cambridge Univ. Press, {\bf 93} (2003).

\bibitem{Gut} J. Gutierrez Garcia, M. A. de Prada Vicente and A. P. \v{S}ostak, {\it A unified approach to the concept of fuzzy $L$-uniform space}, Chapter 3 in: S. E. Rodabaugh, E. P. Klement, eds, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic Kluwer Academic Publishers (Boston/Dordrecht/London), {\bf20} (2003), 81--114.

\bibitem{Hof} K. H. Hofmann, J. D. Lawson, J. S. Pym and eds., {\it The analytical and topological theory of semigroups: trends
and developments}, Berlin, New York: de Gruyter, 1990.

\bibitem{H2} U. H\"{o}hle, {\it Probabilistic uniformization of fuzzy topologies}, Fuzzy Sets and Systems, {\bf1} (1978), 311--332.

\bibitem{Ho} M. Horry and M. M. Zahedi, {\it Uniform and semi-uniform topology on general}, Iranian Journal of Fuzzy Systems, {\bf6} (2009), 19--29.

\bibitem{Hutton} B. Hutton, {\it Uniformities on fuzzy topological spaces}, J. Math. Anal. Appl., {\bf58} (1977), 557--571.

\bibitem{Kat} A. K. Katsaras, {\it On fuzzy uniform spaces}, J. Math. Anal. Appl., {\bf101} (1984), 97--114.

\bibitem{Kot} W. Kotz\'{e}, {\it Uniform spaces}, Chapter 8 in: U. H\"{o}hle, S. E. Rodabaugh and eds, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, (1999), 553--580.

\bibitem{Lowen} R. Lowen, {\it Fuzzy uniform spaces}, J. Math. Anal. Appl., {\bf82} (1981), 370--385.

\bibitem{R1} S. E. Rodabaugh, {\it A theory of fuzzy uniformities with applications to the fuzzy real lines}, J. Math. Anal. Appl., {\bf129} (1988), 37--70.
\bibitem{R3} S. E. Rodabaugh, {\it Axiomatic foundations for uniform operator quasi-uniformities}, Chapter 7 in [4], 199--234.

\bibitem{Shi1} F. G. Shi, {\it Pointwise uniformities in fuzzy set theory}, Fuzzy Sets and Systems, {\bf98} (1998), 141--146.

\bibitem{Shi2} F. G. Shi, {\it Theory and application of pointwise uniformities and metrics on lattices}, Ph.D Thesis, 2001.

\bibitem{Shi3} F. G. Shi, {\it Pointwise pseudo-metric on the $L$-real line}, Iranian Journal of Fuzzy Systems, {\bf2} (2005), 15--20.

\bibitem{Yao} W. Yao, {\it Analogizing Hutton's quasi-uniformities for complete lattices and extending Shi's quasi-uniformities to closed set lattices}, Fuzzy Sets and Systems, {\bf160} (2009), 1233--1244.

\bibitem{Yue} Y. Yue and J. Fang, {\it Extension of Shi's quasi-uniformities in a Kubiak-\v{S}ostak sense}, Fuzzy Sets and Systems, {\bf157} (2006), 1956--1969.
\bibitem{Zhang1} D. Zhang, {\it Stratified Hutton uniformities}, Fuzzy Sets and Systems, {\bf140} (2003), 399--423.

\bibitem{Zhang2} D. Zhang, {\it A comparison of various uniformities in fuzzy topology}, Fuzzy Sets and Systems, {\bf140} (2003), 399--423.