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

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.