On initial and final L-topological groups

In this paper a notion of L-topological group, introduced by Ahsanullah in 1984, is studied. Some basic properties related to these L-topological groups are proved. In 1992 Kubiak had generalized the functors ω and ι, defined by Lowen in 1976, for any complete lattice L to the functors ωL and ιL. We show here that for this notion of L-topological group ωL and ιL 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.