The aim of the present study is to introduce the concept of soft topological transformation groups by examining the topological transformation groups, which are the core subject of algebraic topology under the soft approach. Actions of soft topological groups on soft topological spaces are studied, and the category of soft topological transformation groups is constructed. Also, a translation and conjugation of the soft topological groups are described. Finally, the definitions of soft orbit spaces and soft homogeneous spaces are given, and some of the properties of these concepts are examined in detail.

This paper introduces the definition of a Lie rough group as a natural development of the concepts of a smooth manifold and a rough group on an approximation space. Furthermore, the properties of Lie rough groups are discussed. It is shown that every Lie rough group is a topological rough group, and that the product of two Lie rough groups is again a Lie rough group. We define the concepts of Lie rough subgroups and Lie rough normal subgroups. Finally, our aim is to give an example by using definition of Lie rough homomorphism sets G and H.

The soft set theory proposed by Molodtsov is a recent mathematical approach for modeling uncertainty and vagueness. The main aim of this study is to introduce the concept of soft action by combining soft set theory with the action which is an important concept in dynamical systems theory. Moreover, di¤erent types of soft action are presented and some important characterizations are given. Finally, we de…ne the concept of soft symmetric group and present the relation between the soft action and soft symmetric group, as a similar result to the classical Cayley's Theorem.

Molodtsov proposed the theory of soft sets which can be considered as a recent mathematical tool to deal with uncertainties. The main purpose of this paper is to give the definition of soft topological hypergrupoid by examining the concept of hypergrupoid which is one of the hyperystructures with soft set theory from the topological point of view. Also, the relation between soft topological hypergroupoids and soft hypergroupoids is examined and some theoretical results are obtained. By introducing the concept of soft good topological homomorphism, the category of soft topological hypergrupoids is constructed. At last, the definition of soft topological subhypergrupoid is presented and some related properties are studied.

The aim of this article is to introduce the concept of an idealistic soft topological hyperring over a hyperring. Some structural properties of this concept are also studied. Moreover, this study investigates the relationship between the idealistic soft topological hyperrings and the idealistic soft hyperrings. Finally, the restricted (extended) intersection and ∧-intersection of the family of the idealistic soft topological hyperrings are examined.