Global L-neighborhood groups

This paper introduces and studies the notion of global L-neighborhood group which is defined as a group equipped with a global L-neighborhood structure in sense of Gähler et al. such that both the binary operation and the unary operation of the inverse are continuous with respect to this global L-neighborhood structure. Some examples of global L-neighborhood groups are given. It is shown that the L-topological groups, given by Ahsanullah in 1984 and later by Bayoumi in 2003, are special global L-neighborhood groups. We also show that all initial and final lifts and hence all initial and final global L-neighborhood groups uniquely exist in the category L-GnghGrp of global L-neighborhood groups. These initial and final global L-neighborhood groups are defined using the initial and final global L-neighborhood structures. Moreover, we show that the L-neighborhood groups, defined by Ahsanullah using the L-neighborhood structures in sense of Lowen, are special global L-neighborhood groups, for L=I is the closed unit interval.