Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids

Mikhail Tkachenko

doi:10.3390/axioms9010023

Abstract

We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

Highlights

The present article is a natural continuation of the study in [1,2], where we considered continuous mappings defined on subspaces of products of topological spaces and established a kind of irreducible factorization of those mappings under quite general assumptions
Our purpose here is to focus on the case when f : S → K is a continuous homomorphism of a submonoid S of a product D = ∏i∈ I Di oftopological monoids or groups
We show in Corollary 4 that every continuous homomorphism defined on a finitely retractable submonoid S of a product D = ∏i∈ I Di of topologized monoids has a finite type provided the homomorphism takes values in a topological NSS group K

Summary

Introduction

The present article is a natural continuation of the study in [1,2], where we considered continuous mappings (homomorphisms) defined on subspaces of products of topological spaces (monoids, groups) and established a kind of irreducible factorization of those mappings (homomorphisms) under quite general assumptions. Let S be a dense submonoid of a product ∏i∈ I Di of semitopological monoids with open shifts and f : S → K be a continuous homomorphism to a Hausdorff paratopological group K.

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Conclusion