The Study of Soft Groups Based on Soft Binary Operations
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Date
2025-06-18
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Addis Ababa University
Abstract
In this thesis, we propose a new de nition for soft groups based on soft binary operations. The
idea is to bring the archetype of 'softness' into the spectrum of algebraic structures using soft
binary operations parametrized by a given set of suitable parameters. One of our achievement
is that we obtain an ordinary group model representing our soft group. The existing classical
group serves as a model to describe and characterize the overall internal properties of our
soft groups. In this vein, we further investigate the soft subgroups (respectively, normal soft
subgroups) and proved some structural theorems.
In this thesis, we also study soft homomorphisms on soft groups and investigate their
properties. Given a soft mapping hf;Ai from G to G0, we obtain an ordinary map
^f
from the
set SEA(G) of soft elements of G to the set SEA(G0) of soft elements of G0, and show that
hf;Ai is a soft homomorphism (respectively, soft isomorphism) if and only if
^f
is an ordinary
group homomorphism (respectively, isomorphism). We apply this concept to study soft
isomorphism theorems on soft groups. In addition, we study those soft automorphisms of soft
groups and the particular class of soft inner automorphisms.
Moreover, we study a few soft group-related ndings based on soft binary operations,
including soft orbits, soft stabilizers, and the action of a soft group on a set. Given a soft
mapping h ;Ai from G X to X; we obtain an ordinary map b from the set
SEA(G) SEA(X) to the set SEA(X) and show that h ;Ai is a soft action if and only if b is
an ordinary action. Finally, we present the fundamental ideas and characteristics of normal
fuzzy soft subgroups.
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Keywords
Study of Soft Groups, Based on Soft Binary, Operations