Studies On Some Properties Of Paratopological Rough Groups

Let（X,·）be a group and（X,τ）be a topological space.If the product map is continuous,then（X,·,τ）is said to be a paratopological group.Inspired by the study for paratopological groups,in this paper,we generalize topological rough groups to paratopological rough groups and study some properties of paratopological rough groups,which is composed of the following two parts.In the first part（Chapter 2）,we study some of the topological properties of paratopological rough groups.It consists of five sections.In the first section,we mainly construct a paratopological rough group which is not a topological rough group.In the second section,we first discuss how to derive the neighborhood base at arbitrary point from the neighborhood base at the rough identity element in paratopological rough groups（Theorem 2.2.4）.And then we discuss the properties of the neighborhood base at the rough identity element in paratopological rough groups（Theorem 2.2.5）and the paratopologized problem of rough groups（Theorem 2.2.6）.In the third section,we mainly study rough subgroups and rough normal subgroups of paratopological rough groups.In particular,we point out that the proof is not true that Lin Fucai et al.use[22,Example 3]to prove that the inverse proposition of[18,Proposition 25]is incorrect.And we construct an example（Example 2.3.4）to show that the inverse proposition of[18,Proposition 25]actually does not hold.In addition,we discuss when the intersection of two rough subgroups of a paratopological rough group is a paratopological rough subgroup（Theorem 2.3.5）.For two rough subgroupsH₁,H₂of a paratopological rough group,we discuss whenH₁H ₂ is a paratopological rough subgroup（Theorem 2.3.6）.For two rough normal subgroupsH₁,H₂of a paratopological rough group,we discuss whenH₁H ₂ is a paratopological rough normal subgroup（Theorem 2.3.8）.In the fourth section,inspired by the rough action on topological rough groups,we introduce the definition of the rough action on paratopological rough groups and explore the homeomorphic properties of two relevant mappings（Theorem 2.4.3）and so on.In the fifth section,we mainly discuss the product of paratopological rough groups,showing that the Tychonoff product of a family of paratopological rough groups is still a paratopological rough group（Theorem 2.5.3）.In the second part（Chapter 3）,we continue to discuss the closure operation of topological rough groups.First,we construct an example（Example 3.1）to clarify some relationships about certain closures in topological rough groups,and then discuss when the closure of a rough subgroup of a topological rough group is a topological rough subgroup and so on.