Stratified lattice-valued balanced neighborhood topological groups (Q2873488)

One of the important characterizations for a~topological group is that the left and the right uniformities coincide if and only if the topological group is balanced. In the sense of the unified approach of lattice-valued uniformities introduced in [\textit{J. Gutiérrez García} et al., in: Topological and algebraic structures in fuzzy sets. A handbook of recent developments in the mathematics of fuzzy sets. Dordrecht: Kluwer Academic Publishers. 81--114 (2003; Zbl 1061.54006)] the left, right, and two-sided stratified lattice-valued uniformities are different in non-abelian groups. The purpose of the paper under review is to introduce the notion ``balanced stratified lattice-valued neighborhood topological group'' for which the left and the right uniformities coincide. The author also introduces the notion of equicontinuity in stratified lattice-valued neighborhood topological spaces and the notion of uniform equicontinuity in stratified lattice-valued uniform spaces. Using these notions he characterizes stratified lattice-valued neighborhood topological groups. He introduces the notions of lattice-valued neighborhood open function and of lattice-valued neighborhood uniformly open function and he shows that they are equivalent in stratified lattice-valued neighborhood topological groups. Finally, the author obtains a characterization of balanced stratified lattice-valued neighborhood topological groups in terms of the uniform continuity and of the binary group operation.