On initial and final L-topological groups

Global -neighborhood groups

“…Hence, the properties and results proved for the global L-neighborhood groups also are true for the L-topological groups. For example, all initial and final lifts and all initial and final L-topological groups uniquely exist in the category L-TopGrp and this means that one of the important results in [4], that the category L-TopGrp is topological over the category Grp of groups, can be also obtained from the results of this paper.…”

“…Definition 2.1 (Ahsanullah [2], Bayoumi [4]). Let G be a group and an L-topology on G. Then (G, ) is called an L-topological group if the mappings…”

“…Denote by L-TopGrp the class of all L-topological groups and the L-continuous homomorphisms between them. It is shown in [4] that L-TopGrp is a topological category over the category Grp of groups and that L-TopGrp also is an algebraic category over the category L-Top of L-topological spaces.…”

“…One of the main results of [4] is to show that the category L-TopGrp of L-topological groups is topological over the category Grp of groups. After the last discussion in this paper the result of [4] can be also obtained as follows.…”

“…In [4] we studied the notion of L-topological group which is defined as a group G equipped with an L-topology on G such that both the binary operation and the unary operation of the inverse are L-continuous with respect to this L-topology. It is shown that the initial and final lifts and the initial and final L-topological groups uniquely exist in the category L-TopGrp of L-topological groups and that the initial and final L-topological groups are defined by means of the initial and final L-topologies.…”