The mapping $i_{2}$ on the free paratopological groups

TL;DR

This paper characterizes when the natural mapping from a product space to the second level of the free paratopological group is closed, linking it to the finest quasi-uniformity on the space.

Contribution

It improves previous results by providing a precise condition for the closedness of the mapping $i_2$ in terms of quasi-uniformities.

Findings

The mapping $i_2$ is closed iff neighborhoods of the diagonal are in the finest quasi-uniformity.

The result applies to $T_1$-spaces with the discrete topology.

Provides a characterization connecting topology and algebraic structure of free paratopological groups.

Abstract

Let $FP(X)$ be the free paratopological group over a topological space $X$. For each non-negative integer $n\in\mathbb{N}$, denote by $FP_{n}(X)$ the subset of $FP(X)$ consisting of all words of reduced length at most $n$, and $i_{n}$ by the natural mapping from $(X\bigoplus X^{-1}\bigoplus\{e\})^{n}$ to $FP_{n}(X)$. In this paper, we mainly improve some results of A.S. Elfard and P. Nickolas's [On the topology of free paratopological groups. II, Topology Appl., 160(2013), 220--229.]. The main result is that the natural mapping $i_{2}: (X\bigoplus X_{d}^{-1}\bigoplus\{e\})^{2}\longrightarrow FP_{2}(X)$ is a closed mapping if and only if every neighborhood $U$ of the diagonal $\Delta_{1}$ in $X_{d}\times X$ is a member of the finest quasi-uniformity on $X$, where $X$ is a $T_{1}$-space and $X_{d}$ denotes $X$ when equipped with the discrete topology in place of its given topology.