Boundedness and Continuity of Fuzzy Linear Order-Homomorphisms on $I$-Topological\\ Vector Spaces

Document Type : Research Paper


1 School of Mathematical Science, Nanjing Normal University, Nan- jing, Jiangsu 210023, P. R. China

2 Department of Mathematics, Anhui NormalUniversity, Wuhu, Anhui 241000, P. R. China


In this paper, a new definition of bounded fuzzy linear order
homomorphism on $I$-topological vector spaces is introduced. This
definition differs from the definition of Fang [The continuity of
fuzzy linear order-homomorphism. J. Fuzzy Math. {\bf
5}\textbf{(4)}(1997), 829--838]. We show that the ``boundedness"
and `` boundedness on each layer" of fuzzy linear order
homomorphisms do not imply each other. On the basis,
characterizations of continuity of fuzzy linear
order-homomorphisms, and the relation between continuity and
boundedness are studied.


\bibitem{Fang1} J. X. Fang, {\it Fuzzy linear order-homomorphism and its
structures}, J. Fuzzy Math., {\bf 4}\textbf{(1)} (1996), 93--102.
\bibitem{Fang2} J. X. Fang, {\it The continuity of fuzzy linear
order-homomorphism}, J. Fuzzy Math., {\bf 5}\textbf{(4)} (1997),
\bibitem{Fang3} J. X. Fang, {\it On local bases of fuzzy topological vector
spaces}, Fuzzy Sets and Systems, {\bf 87} (1997), 341--347.
\bibitem{HR} U. H\"ohle and S. E. Rodabaugh, eds., {\it Mathematics
of fuzzy sets: logic, topology, and measure theory}, The Handbooks
of Fuzzy Sets Series, Kluwer Academic Publishers,
Dordrecht, {\bf 3} (1999).
\bibitem{JY} S. Q. Jiang and C. H. Yan, {\it Fuzzy bounded sets and
totally fuzzy bounded sets in $I$-topological vector spaces},
Iranian Journal of Fuzzy Systems, {\bf 6}\textbf{(3)} (2009), 73--90.
\bibitem{KL} A. K. Katsaras and D. B. Liu, {\it Fuzzy vector spaces and fuzzy
topological vector spaces}, J. Math. Anal. Appl., {\bf 58} (1977),
\bibitem{Ka1} A. K. Katsaras, {\it Fuzzy topological vector spaces I}, Fuzzy
Sets and Systems, {\bf 6} (1981), 85--95.
\bibitem{Ka2} A. K. Katsaras, {\it Fuzzy topological vector spaces II}, Fuzzy
Sets and Systems, {\bf 12} (1984), 143--154.
\bibitem{Lo} R. Lowen, {\it Fuzzy topological spaces and fuzzy
compactness}, J. Math. Anal. Appl., {\bf 56} (1976), 621--633.
\bibitem{PL} P. M. Liu, {\it Fuzzy topology I, neighborhood
structures of a fuzzy points and Moore-Smith convergence}, J.
Math. Anal. Appl., {\bf 76} (1980), 571--599.
\bibitem{Ro1} S. E. Rodabaugh, {\it Point-set lattice-theoretic
topology}, Fuzzy Sets and Systems, {\bf 40} (1991), 297--347.
\bibitem{Ro2} S. E. Rodabaugh, {\it Powerset operator based foundation for
point-set lattice-theoretic (POSLAT) fuzzy set theories and
topologies}, Quaestiones Mathematicae, {\bf 20} (1997), 463--530.
\bibitem{Wang} G. J. Wang, {\it Order-homomorphisms of fuzzes}, Fuzzy Sets and
Systems, {\bf 12} (1984), 281--288.
\bibitem{Wa} R. H. Warren, {\it Neighborhoods, bases and continuity in fuzzy
topological spaces}, Rocky Mountain J. Math., {\bf 8} (1978),
\bibitem{WF} C. X. Wu and J. X. Fang, {\it Boundedness and locally bounded
fuzzy topological vector spaces}, Fuzzy Math. (China), (in Chinese), {\bf
5}\textbf{(4)} (1985), 87--94.
\bibitem{ZF} H. P. Zhang and J. X. Fang, {\it A note on locally bounded
$L$-topological vector spaces}, Information Sciences, {\bf 179}
(2009), 1792--1794.