Topological Modules of Bounded Bigroup Homomorphisms on a Topological Module

TL;DR

This paper introduces three classes of bounded bigroup homomorphisms between topological modules over a ring, analyzing their algebraic and topological properties including continuity and completeness.

Contribution

It defines new classes of bounded homomorphisms with respect to different uniform convergence topologies and studies their algebraic and topological properties.

Findings

Operations are continuous for each class.

Each class's uniform completeness is investigated.

Provides a framework for bounded homomorphisms in topological modules.

Abstract

Let X, Y, and Z be topological modules over a topological ring R. In this paper, we introduce three different classes of bounded bigroup homomorphisms from X \times Y into Z with respect to the three different uniform convergence topologies. We show that the operations of addition and module multiplication are continuous for each class of bounded bigroup homomorphisms. Also, we investigate whether each class of bounded bigroup homomorphisms is uniformly complete.