We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each $2$-pseudocompact paratopological group is feebly compact and that each Hausdorff $\sigma$-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topologically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a $T_0$ sequentially compact group, a $T_1$ $2$-pseudocompact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free abelian countably compact topological group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We investigate cone topologies of paratopological groups which provide a general tool to construct pathological examples, especially examples of compact-like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup $S$ of a group $G$ and compact-like properties of two basic semigroup topologies generated by $S$ on the group $G$. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group $G$ is feebly compact provided it has a feebly compact normal subgroup $H$ such that a quotient group $G/H$ is feebly compact.

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关于弱紧的副拓扑群

我们获得了许多结果并解决了一些关于弱紧的副拓扑群的问题。我们得到这样一个群是拓扑的充要条件。其中之一是准正则性。我们证明每个$2$-pseudocompact paratopological group 是弱紧的，并且每个Hausdorff $\sigma$-compact 弱紧并拓扑群是一个紧拓扑群。我们特别关注周期性和拓扑周期性群。我们构造了各种非拓扑群的类紧致并拓扑群的例子，其中包括一个 $T_0$ 顺序紧致群，一个 $T_1$ $2$-伪紧致群，一个泛函 Hausdorff 可数紧群（在公理假设下有一个无限无扭阿贝尔可数紧拓扑群，没有非平凡收敛序列），以及一个泛函 Hausdorff 第二个可数群序序紧实群。我们研究了副拓扑群的锥拓扑，它提供了构建病理示例的通用工具，尤其是具有不连续反转的紧凑型副拓扑群的示例。我们发现群 $G$ 的基本锥子半群 $S$ 的代数性质与群 $G$ 上由 $S$ 生成的两个基本半群拓扑的类紧致性质之间存在简单的相互作用。我们证明弱紧副拓扑群族的乘积是弱紧的，