Soft Set and Soft Group from Classical View Point

Abstract

It is shown that a soft set can be represented as a crisp set of soft elements and a soft group as a ordinary group of soft elements. From this view point it is immediate that soft group share the properties of ordinary group. Also using soft elements the definitions of soft co-sets, soft homomorphism and cyclic soft groups are presented and their properties are studied.

Soft Bornological Group Acts on Soft Bornological Set

“…Definition 2.4 [10]: Let 𝐺 be a group and 𝐴 be a nonempty subset of parameters𝐸. For the soft set (𝜉, 𝐴) over 𝐺, (𝜉, 𝐴) is said to be a soft group over 𝐺 if and only if 𝜉(𝑒) ≤ 𝐺, for all 𝑒 ∈ 𝐴.…”

“…Many researchers have contributed to developing soft set theory's algebraic structure to the best. For more details see [6] [7] [8] [9] [10]. To tackle the problems of boundedness for a soft set, one can merge the soft set with bornology to develop a new structure called the soft bornological set [11].…”

On Soft Lebesgue Measure

“…Definition 2.3. [51] Let F be a soft set then the collection of all soft elements of F is denoted by SE(F ). That is SE(F ) = {a : a : A → X, a(t) ∈ F (t), ∀ t ∈ A }.…”

On Soft Lebesgue Measure

“…Thus the idea of soft set is more general than the idea of fuzzy set. Definition 2.9 [38]. A function a : A → X is called soft element of a soft set F if a(t) ∈ F (t) for all t ∈ A and F (t) ̸ = ϕ for all t ∈ A..…”

“…In this case we write a ∈ s F . Definition 2.10 [38]. Let F be a soft set then the collection of all soft elements of F is denoted by SE(F ).…”