Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids

Mikhail Tkachenko

doi:10.3390/axioms8030086

Abstract

We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = ∏ i ∈ I D i of topologized monoids and f : S → H is a continuous homomorphism to a topologized semigroup H with ψ ( H ) ≤ ω , then one can find a countable subset E of I and a continuous homomorphism g : p E ( S ) → H satisfying f = g ∘ p E ↾ S , where p E is the projection of D to ∏ i ∈ E D i . The same conclusion is valid if S contains the Σ -product Σ D ⊂ D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E ⊂ I with the aforementioned properties.

Highlights

The present article is a natural continuation of [1], where we study continuous mappings of products of topological spaces and establish the existence of an irreducible factorization of those mappings under quite general assumptions
Our purpose here is to consider the case when f : S → H is a continuous homomorphism of a submonoid S of a product D = ∏i∈ I Di of topologized monoids
In Corollaries 1 and 2 we show that Proposition 1 is valid for a submonoid S of the product D = ∏i∈ I Di of topologized monoids Di provided that S is either ω-retractable or contains the Σ-product ΣD ⊂ D

Summary

Introduction

The present article is a natural continuation of [1], where we study continuous mappings (of subspaces) of products of topological spaces and establish the existence of an irreducible factorization of those mappings under quite general assumptions. Our purpose here is to consider the case when f : S → H is a continuous homomorphism of a submonoid S of a product D = ∏i∈ I Di of topologized monoids.

Results

Conclusion