Enriched Lattice-Valued Topological Groups

Abstract

In this paper, we focus on enriched cl-premonoid-valued topological groups, and their so-called change-of-basis lattice. In so doing, we take L as an enriched cl-premonoid and present a category SL-NTopGrp, of strati¯ed enriched cl-premonoid-valued neighborhood topological groups. We produce some characterization theorems, and prove that every strati¯ed L-neighborhood topological group is uniformizable. Finally, we look at the enriched lattice-valued neighborhood topological group when the underlying basis is changed under certain functorial mechanism.

Lattice-valued quasi-bi-uniformizability of lattice-valued quasi-bi-topological neighborhood groups

“…Then it has always three L-quasi-uniformities: U L (the left L-quasi-uniformity), U R (the right L-quasi-uniformity) and U B (the two-sided L-quasi-uniformity, Proposition 3.5 [4]). Having the same notations as in Section 3 [4] (see also Section 4 [3] with L, a frame), one obtains the following Theorem 4.7. Each stratified L-quasi-bi-topological neighborhood group…”

“…Proof. If (X, •, Ψ) is a stratified L-topological neighborhood group, then it follows from Theorem 4.2 [3] that U L , U R and U B always exist. To prove the converse, we consider the case for U L , assuming that (X, U L ) is weakly locally symmetric.…”

“…Generalization of classical notion of topological groups into the framework of lattice-valued case have received much attention over the years [3,4,5,7,17], we provide a short list of reference which by no means is complete (for an almost complete list, we refer the interested reader to [3]). However, we do not see much work on the generalization of classical notion of quasi-topological groups (which is also widely known as paratopological groups in the sense of Bourbaki [6]) within the scope of lattice-valued mathematics.…”

“…Considering a few results of classical quasi-topological groups, in this article, with L as a frame, and the same stratified L-topological neighborhood system used in [3,4,5] as originated from [13], we introduce the notion of stratified L-quasi-topological neighborhood group, stratified semi-topological neighborhood group and stratified L-quasi-bi-topological neighborhood group.…”

On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

“…We have investigated a notion of L-valued topological groups in [3], where we considered L-valued subgroup of a group. Various aspects of L-valued subgroups of groups are studied over the years by various authors, cf.…”