Free Paratopological Groups Research Articles

A note on paratopological groups with an ωω-base

In this paper, paratopological groups with an ωω-base are investigated. The following results are obtained, which generalizes some conclusions in literature. (1) Every Fréchet-Urysohn Hausdorff paratopological group having the property (**) with an ωω-base is first-countable, hence submetrizable, where a paratopological group G has the property (**) if there exist a non-trivial sequence {xn}n∈N in G such that both {xn}n∈N and {xn−1}n∈N converge to the identity of G. (2) The free Abelian paratopological group AP(X) on a topological space X has an ωω-base if and only if the fine quasi-uniformity of X has an ωω-base. (3) If X is a countable topological space, then the free paratopological group FP(X) on X has an ωω-base if and only if the fine quasi-uniformity of X has an ωω-base.

Characterizing s-paratopological groups by free paratopological groups

A paratopological group G is called an s-paratopological group if every sequentially continuous homomorphism from G to a paratopological group is continuous. In this paper, the structure of s-paratopological groups is established in terms of free paratopological groups. Namely, if G is a non-discrete T1 paratopological group, then the following statements (1), (2), (3) and (4) are equivalent. (1) G is an s-paratopological group. (2) G is topologically isomorphic to a quotient group of a free paratopological group on a metrizable space. (3) G is topologically isomorphic to a quotient group of a free paratopological group on a T1 Fréchet space. (4) G is topologically isomorphic to a quotient group of a free paratopological group on a T1 sequential space.

The countable type properties in free paratopological groups

A space X is of countable type (resp. subcountable type) if every compact subspace F of X is contained in a compact subspace K that is of countable character (resp. countable pseudocharacter) in X. In this paper, we mainly show that: (1) For a functionally Hausdorff space X, the free paratopological group FP(X)and the free abelian paratopological group AP(X) are of countable type if and only if X is discrete; (2) For a functionally Hausdorff space X, if the free abelian paratopological group AP(X) is of subcountable type then X has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff μ-space X, if AP2(X) or FP2(X) is locally compact, then X is a topological sum of a compact space and a discrete space.

Mapping i2 on the free paratopological groups

Let FP(X) be the free paratopological group over a topological space X. For each nonnegative integer n ? N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and in by the natural mapping from (X ? X?1 ? {e})n to FPn(X). We prove that the natural mapping i2:(X ? X?1 d ?{e})2 ? FP2(X) is a closed mapping if and only if every neighborhood U of the diagonal ?1 in Xd x X is a member of the finest quasi-uniformity on X, where X is a T1-space and Xd denotes X when equipped with the discrete topology in place of its given topology.

MP-equivalence of free paratopological groups

Let FP(X) denote the free paratopological group over a topological space X. Two topological spaces X and Y are called MP-equivalent if FP(X) and FP(Y) are topologically isomorphic. At first, it is shown that there exist non-homeomorphic topological spaces X and Y such that FP(X) and FP(Y) are topologically isomorphic. Secondly, MP-invariance of free paratopological groups is investigated. It is established that pseudocompactness, hereditary Lindelöfness, hereditary separability and the property of being a cosmic space are all MP-invariant, which generalizes some conclusions valid for free topological groups to free paratopological groups. Finally, a few questions about MP-equivalence of free paratopological groups are posed.

A few generalized metric properties on free paratopological groups

In this paper, some generalized metric properties on free paratopological groups are obtained. At first, a general stability theorem of free paratopological groups on submetrizable spaces is established. Secondly, countable tightness of free paratopological groups on k⁎-metrizable k-spaces or k-semistratifiable k-spaces is investigated. Finally, first-countability of free paratopological groups is discussed and strongly Fréchetness of free paratopological groups is characterized. These complement and improve some known conclusions in literature. In addition, some questions are posed.

On local compactness of the subspace FP2(X) of FP(X)

In this paper, it is shown that if X is homeomorphic to the topological sum of a Tychonoff space Y and a discrete space D, then FP2(X) is locally compact if and only if FP2(Y) is locally compact. Especially, the space FP2(X) is locally compact at its every point distinct from the identity e if X is homeomorphic to the topological sum of a compact space and a discrete space. Finally, a few questions about free paratopological groups are posed.

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.

Free Abelian paratopological groups over metric spaces

Let X be a metrizable space. Let FP(Y) and AP(X) be the free paratopological group over X and the free Abelian paratopological group over X, respectively. Firstly, we use asymmetric locally convex spaces to prove that if Y is a subspace of X then AP(Y) is topological subgroup of AP(X). Then, we mainly prove that:(a)if the tightness of AP(X) is countable then the set of all non-isolated points in X is separable;(b)if X is a z-space, then AP(X) is a k-space if and only if X is locally compact, locally countable and the set of all non-isolated points in X is countable;(c)AP2(X) is first-countable if and only if the set of all non-isolated points in X is finite.Moreover, we show that, for a Tychonoff space X, AP(X) has a countable k-network if and only if X is a countable space with a countable k-network. Finally, we give negative answers to three questions which were posed by Arhangel'skiı̌ and Tkachenko in [3]. Some questions concerned with free paratopological groups are posed.

Neighborhood base at the identity of free paratopological groups

AbstractIn 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.

On the topology of free paratopological groups. II

Let FP(X) be the free paratopological group on a topological space X. For n∈N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and by in the natural mapping from (X⊕X−1⊕{e})n to FPn(X). In this paper a neighbourhood base at the identity e in FP2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i2:(X⊕Xd−1⊕{e})2→FP2(X) is a quotient mapping, where X is a T1 space and Xd−1 denotes the set X−1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T1 space. Several specific spaces and classes of spaces are also examined. For example, i2 is a quotient mapping for every countable subspace of R, i2 is not a quotient mapping for any uncountable compact subspace of R, and it is undecidable in ZFC whether an uncountable subspace of R exists for which i2 is a quotient mapping.

A note on free paratopological groups

In this paper, we mainly discuss some generalized metric properties and the character of the free paratopological groups, and extend several results valid for free topological groups to free paratopological groups.

Topological monomorphisms between free paratopological groups

Suppose that $X$ is a subspace of a Tychonoff space $Y$. Then the embedding mapping $e_{X, Y}: X\rightarrow Y$ can be extended to a continuous monomorphism $\hat{e}_{X, Y}: AP(X)\rightarrow AP(Y)$, where $AP(X)$ and $AP(Y)$ are the free Abelian paratopological groups over $X$ and $Y$, respectively. In this paper, we mainly discuss when $\hat{e}_{X, Y}$ is a topological monomorphism, that is, when $\hat{e}_{X, Y}$ is a topological embedding of $AP(X)$ to $AP(Y)$.

On the topology of free paratopological groups

The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group $F(X)$ on a Tychonoff space$X$. In this paper, an analogue of Joiner's lemma for the free paratopological group $\FP(X)$ on a $T_1$ space $X$ is proved. Using this, it is shown that the following conditions are equivalent for a space $X$: (1) $X$ is $T_1$; (2) $\FP(X)$ is $T_1$; (3) the subspace $X$ of $\FP(X)$ is closed; (4) the subspace $X^{-1}$ of $\FP(X)$ is discrete; (5) the subspace $X^{-1}$ is $T_1$; (6) the subspace $X^{-1}$ is closed; and (7) the subspace $\FP_n(X)$ is closed for all $n \in \N$, where $\FP_n(X)$ denotes the subspace of $\FP(X)$ consisting of all words of length at most $n$.

Free paratopological groups and free products of paratopological groups

We study the topological and algebraic properties of free paratopological groups and free products of paratopological groups.