Global [formula omitted]-neighborhood groups

Abstract

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.