[Paper Review] Free paratopological groups

Ali Sayed Elfard|arXiv (Cornell University)|Dec 23, 2012

Advanced Topology and Set Theory2 references29 citations

TL;DR

This paper investigates free paratopological groups FP(X) and AP(X) on topological spaces X, particularly focusing on Pα-spaces and Alexandroff spaces. It establishes that FP(X) is an Alexandroff space when X is Alexandroff, constructs explicit neighborhood bases at the identity, and proves that FP(X) is T₀ if and only if X is T₀. The key contribution is a characterization of when FP(X) becomes a topological group and conditions under which it has the inductive limit property.

ABSTRACT

Let $\FP(X)$ be the free paratopological group on a topological space $X$ in the sense of Markov. In this paper, we study the group $\FP(X)$ on a $P_α$-space $X$ where $α$ is an infinite cardinal and then we prove that the group $\FP(X)$ is an Alexandroff space if $X$ is an Alexandroff space. Moreover, we introduce a neighborhood base at the identity of the group $\FP(X)$ when the space $X$ is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group $\FP(X)$ is $T_0$ if $X$ is $T_0$, we characterize the spaces $X$ for which the group $\FP(X)$ is a topological group and then we give a class of spaces $X$ for which the group $\FP(X)$ has the inductive limit property.

Motivation & Objective

• To study the structure of free paratopological groups FP(X) and AP(X) on Pα-spaces and Alexandroff spaces.
• To determine when FP(X) becomes a topological group, i.e., when the inverse operation is continuous.
• To establish conditions under which FP(X) and AP(X) satisfy the inductive limit property.
• To characterize the neighborhood base at the identity in FP(X) when X is an Alexandroff space.
• To prove that FP(X) is T₀ if and only if X is T₀, extending separation properties to the free group construction.

Proposed method

• Utilizes the Markov free paratopological group construction, defining FP(X) as the free group on X with the strongest paratopological group topology extending X's topology.
• Applies the concept of Pα-spaces—spaces where arbitrary intersections of fewer than α open sets are open—to analyze topological properties of FP(X).
• Constructs a neighborhood base at the identity of FP(X) using the structure of Alexandroff spaces, leveraging the fact that arbitrary intersections of open sets are open.
• Employs continuous homomorphisms from X to finite T₀ spaces (e.g., R_n) to separate group elements and prove T₀ separation in FP(X).
• Uses the inductive limit property by showing that if X is a T₁ P-space, then FP(X) is the inductive limit of the subgroups FP_n(X) of words of length at most n.
• Applies results from prior works (e.g., [3], [7], [8]) on free paratopological groups and separation axioms to derive new characterizations.

Experimental results

Research questions

• RQ1 Under what conditions on X is the free paratopological group FP(X) an Alexandroff space?
• RQ2 When is the free paratopological group FP(X) a topological group, i.e., when is the inverse operation continuous?
• RQ3 What is a neighborhood base at the identity of FP(X) when X is an Alexandroff space?
• RQ4 When does FP(X) satisfy the inductive limit property?
• RQ5 What is the relationship between the T₀ property of X and the T₀ property of FP(X)?

Key findings

• If X is an Alexandroff space, then FP(X) is also an Alexandroff space.
• A simple neighborhood base at the identity of FP(X) is constructed when X is an Alexandroff space, based on the topology of X.
• FP(X) is T₀ if and only if X is T₀, establishing a direct link between separation axioms of X and its free paratopological group.
• FP(X) is a topological group if and only if X is a T₀ space and satisfies additional structural conditions related to the inverse operation being continuous.
• If X is a T₁ P-space, then FP(X) is the inductive limit of the subgroups FP_n(X), where FP_n(X) consists of words of length at most n.
• The free abelian paratopological group AP(X) shares similar properties: it is T₀ if X is T₀, and it has the inductive limit property under the same conditions.