# Boundedness of linear order-homomorphisms in \$L\$-topological vector spaces

Document Type: Research Paper

Authors

1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

2 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Abstract

A new definition of boundedness of linear order-homomorphisms (LOH)
in \$L\$-topological vector spaces is proposed. The new definition is
compared with the previous one given by Fang [The continuity of
fuzzy linear order-homomorphism, J. Fuzzy Math. 5 (4) (1997)
829\$-\$838]. In addition, the relationship between boundedness and
continuity of LOHs is discussed. Finally, a new uniform boundedness
principle in \$L\$-topological vector spaces is established in the
sense of a new definition of uniform boundedness for a family of
LOHs.

Keywords

### References

[1] D. Dubois and H. Prade, Fuzzy sets and systems: theory and applications, Academic Press,
New York, 1980.
[2] M. A. Erceg, Functions, equivalence relations, quotient spaces and subsets in fuzzy set theory,
Fuzzy Sets and Systems, 3 (1980), 75{92.
[3] J. X. Fang, Fuzzy linear order-homomorphism and its structures, J. Fuzzy Math., 4(1) (1996),
93{102.
[4] J. X. Fang, The continuity of fuzzy linear order-homomorphism, J. Fuzzy Math., 5(4) (1997),
829{838.
[5] J. X. Fang and C. H. Yan, L-fuzzy topological vector spaces, J. Fuzzy Math., 5(1) (1997),
133{144.

[6] J. X. Fang and H. Zhang, Boundedness and continuity of fuzzy linear order-homomorphisms
on I-topological vector spaces, Iranian Journal of Fuzzy Systems, 11(1) (2014), 147{157.
[7] M. He, Bi-induced mappings on L-fuzzy sets, Kexue Tongbao, (in Chinese), 31 (1986), 475.
[8] U. Hohle and S. E. Rodabaugh (Eds.), Mathematics of fuzzy sets: logic, topology and measure
theory, the handbooks of fuzzy sets series, vol. 3, Kluwer Academic Publishers, Dordrecht,
1999.
[9] A. K. Katsaras, Fuzzy topological vector spaces I, Fuzzy Sets and Systems, 6 (1981), 85{95.
[10] Y. M. Liu, Structures of fuzzy order homomorphisms, Fuzzy Sets and Systems, 21 (1987),
43{51.
[11] Y. M. Liu and M. K. Luo, Fuzzy topology, World Scienti c Publishing, Singapore, 1997.
[12] S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems, 40 (1991),
297{345.
[13] S. E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic
(POSLAT) fuzzy set theories and topologies, Quaestiones Math., 20 (1997), 463{530.
[14] G. J. Wang, Order-homomorphisms on fuzzes, Fuzzy Sets and Systems, 12 (1984), 281{288.
[15] G. J. Wang, Theory of L-fuzzy topological spaces, Shaanxi Normal University Press, Xi'an,
(in Chinese), 1988 .
[16] C. H. Yan, Initial L-fuzzy topologies determined by the family of L-fuzzy linear order-
homomorphisms, Fuzzy Sets and Systems, 116 (2000), 409{413.
[17] C. H. Yan, Generalization of inductive topologies to L-topological vector spaces, Fuzzy Sets
and Systems, 131 (2002), 347{352.
[18] C. H. Yan and J. X. Fang, The uniform boundedness principle in L-topological vector spaces,
Fuzzy Sets and Systems, 136 (2003), 121{126.