Isotopic Properties of Neutrosophic Soft Quasigroup and Its Application in Decision-making

A Q-neutrosophic soft quasigroup (ϕ Q, A) represents a novel mathematical framework designed to address scenarios characterized by indeterminate occurrences. This paper seeks to extend the classical examination of quasigroups to include Q-neutrosophic soft quasigroups. It provides a thorough investigation into isotopism, homomorphism, isomorphism, and the direct product concerning the hybrid concept of neutrosophic soft sets under a non-associative structure. This research expands the concept of the Q-set to a groupoid, which facilitates the development of an algorithm for Q-neutrosophic soft quasigroups. The definitions of isotopism and homomorphism for Q-neutrosophic soft quasigroups are introduced, revealing that every isotope of a Qneutrosophic soft quasigroup is indeed a Q-neutrosophic soft quasigroup. Furthermore, it is established that the homomorphic image of a Q-neutrosophic soft quasigroup under a quasigroup is not necessarily a Q-neutrosophic soft quasigroup. The findings indicate that the direct product of any two Q-neutrosophic soft quasigroups is also a Q-neutrosophic soft quasigroup. Generalized properties regarding the direct product of Q-neutrosophic soft quasigroups of order n have been established. Additionally, the study presents the necessary and su cient conditions under which the direct product of two Q-neutrosophic soft subquasigroups, ψ Q(a)×ψ Q(b), contains at least one Q-neutrosophic soft subquasigroup, πQ(c), that is isomorphic to the Q-neutrosophic soft subquasigroup πQ(a). Specifically, ψ Q(a) ∼= πQ(c) ≤ ψ Q(a)×ψ Q(b) if and only if a homomorphism exists from ψ Q(a) to ψ Q(b). To validate some of these results, examples have been constructed, and a practical application utilizing real-life data has been demonstrated.