Free paratopological groups

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 $P_\alpha$-space $X$ where $\alpha$ 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.

💡 Research Summary

The paper investigates the free paratopological group FP(X) (and its abelian counterpart AP(X)) on a topological space X, focusing on the interplay between the algebraic freeness and the underlying topology. The authors introduce the notion of a Pα‑space for an infinite cardinal α: a space in which the intersection of any family of fewer than α open sets is again open. They define the τ_α‑modification of a given topology τ as the smallest topology containing τ that makes the space a Pα‑space, and they prove (Theorem 3.1) that a base for τ_α consists precisely of intersections of fewer than β open sets, where β equals α if α is regular and α⁺ if α is singular.

Using this framework, the authors show (Proposition 3.3) that the free paratopological group FP(X) (and AP(X)) is a Pα‑space if and only if X itself is a Pα‑space. Since an Alexandroff space is exactly a space that is a Pα‑space for every infinite α, they obtain the central result (Theorem 4.1): FP(X) and AP(X) are Alexandroff spaces precisely when X is Alexandroff.

A major technical contribution is the explicit construction of a neighborhood base at the identity element. For each point x∈X, let U(x) be the intersection of all open neighborhoods of x. Define U_F = ⋃{x∈X} x^{-1}U(x) and U_A = ⋃{x∈X} (U(x)−x). The smallest normal submonoid containing U_F (resp. U_A) is denoted N_F (resp. N_A). The authors prove (Theorem 4.4) that the singleton families {N_F} and {N_A} satisfy the four Marin‑Romaguera conditions, thus forming a neighborhood base at the identity for the free (abelian) paratopological group topology. Moreover, they show that the topology generated by these bases coincides with the free topology T_{FP} (resp. T_{AP}), confirming that the constructed bases are indeed the canonical ones.

Armed with these bases, several applications are derived:

1. Topological Group Characterization (Theorem 5.1). FP(X) (or AP(X)) is a genuine topological group (i.e., inversion is continuous) if and only if X is a partition space—equivalently, the family {U(x)}_{x∈X} forms a partition of X. In this case the normal submonoid N_F becomes a subgroup, guaranteeing continuity of inversion.

2. T₀‑Property (Theorem 5.6). If X is T₀, then FP(X) is also T₀. Conversely, FP(X) fails to be T₀ precisely when there exists a non‑trivial element w∈N_F with trivial image under the canonical homomorphism Ĩ: FP(X)→AP(X). This links the T₀‑condition directly to the structure of N_F.

3. Inductive Limit Property (Theorem 5.7). For a class of spaces X (including certain partition spaces), the groups FP(X) and AP(X) possess the inductive limit property: they can be expressed as the direct limit of an increasing chain of subgroups, each equipped with the subspace topology inherited from the free group. This provides a convenient way to analyze their global topology via simpler pieces.

Additional results include Theorem 4.5, which identifies N_F and N_A with the smallest normal subgroups generated by the sets Z_F = {x^{-1}x | x∈X} and Z_A = {x−x | x∈X} when X is indiscrete, and Theorem 4.6, which shows that if X is a partition space then both FP(X) and AP(X) are themselves partition spaces.

Overall, the paper blends set‑theoretic topology (through Pα‑spaces) with algebraic topology of free groups, delivering a clear picture of when free paratopological groups inherit desirable separation, continuity, and limit properties from their base spaces. The explicit neighborhood bases at the identity are particularly valuable, as they enable concrete verification of topological group axioms and facilitate further structural investigations.