In this paper a notion of L-topological group, introduced byAhsanullah in 1984, is studied. Some basic properties related to these L-topological groups are proved. In 1992 Kubiak had generalized the functors
and , defined by Lowenin1976,foranycompletelatticeLtothefunctors
L andL.WeshowherethatforthisnotionofL-topological group
L and L are functors. Moreover, to justify this notion of L-topological group, we show in this paper that all initial and final lifts exist uniquely in the concrete category L-TopGrp of L-topological groups and hence all initial and final L-topological groups exist and can be characterized.As consequences the L-topological subgroups, L-topological product groups, and L-topological quotient groups are exist. Some examples of L-topological groups are given.