Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids

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.

Keywords

• monoid
• homomorphism
• character
• factorization