$G$-compactness for topological groups with operations

TL;DR

This paper explores the concept of G-compactness in various topological algebraic structures, extending previous notions from topological groups to broader contexts like rings, modules, and algebras.

Contribution

It introduces and proves results on different types of G-compactness for topological groups with operations, broadening the scope of the concept.

Findings

Established new results on G-compactness in topological rings and modules.

Extended G-compactness concepts to Lie and Jordan algebras.

Unified framework for G-compactness across multiple algebraic structures.

Abstract

It is well known that for a Hausdorff topological group $X$, the limits of convergent sequences in $X$ define a function denoted by $\lim$ from the set of all convergent sequences in $X$ to $X$. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of $G$-continuity, $G$-compactness and $G$-connectedness. In this paper we prove some results on different types of $G$-compactness for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.