l-fuzzifying topological groups

the main purpose of this paper is to introduce a concept of$l$-fuzzifying topological groups (here $l$ is a completelydistributive lattice) and discuss some of their basic properties andthe structures. we prove that its corresponding $l$-fuzzifyingneighborhood structure is translation invariant. a characterizationof such topological groups in terms of the corresponding$l$-fuzzifying neighborhood structure of the unit is given. it isshown that the category of $l$-fuzzifying topological groups$l$-{bf fytpg} is topological over the category of groups {bf grp}with respect to the forgetful functor. as an application, theconclusion that the product of $l$-fuzzifying topological groups isalso an $l$-fuzzifying topological group is proved. finally, it isproved the forgetful functor preserves the product.