A category of complete residuated lattice-value neighborhood groups

In this paper, considering L a complete residuated lattice, we present a lattice-valued category TNG (resp., TTNG) of (topological) ⊤-neighborhood groups, where the object is defined as a group equipped with a (topological) ⊤-neighborhood space such that the group operations are continuous with respect to the (topological) ⊤-neighborhood space. It is proved that: (1) The ⊤-neighborhood space associated with a ⊤-neighborhood group is topological, so the category TNG is equivalent to the category TTNG. Hence TTNG is redundant, and we need only discuss TNG. (2) The category TNG has nice characterizations, localization and uniformization. (3) The category TNG has initial structure, so it is a topological category, and each initial structure has an ordered representation. (4) The category NG of neighborhood groups can be embedded in TNG as a reflective subcategory. (5) The category TNG can be embedded in the category SLNG of stratified L-neighborhood groups as a reflective subcategory when the underlying lattice L is a meet-continuous lattice. (6) The category TNG is equivalent to the category StrLTOPG of strong L-topological groups when L is a complete MV-algebra.