The rhotrix, introduced as a rhomboidal extension of traditional matrix theory, offers a unique algebraic framework for the organization and manipulation of numerical arrays. The incorporation of refined neutrosophic concepts into rhotrix theory has enabled the modeling of uncertainty, indeterminacy, and inconsistency within algebraic frameworks. Motivated by these developments, this paper investigates the structure of invertible refined neutrosophic rhotrices, thereby extending the classical theory of rhotrices into the domain of refined neutrosophic. This study is carried out using heart-based multiplication, which acts as the fundamental operation governing interactions among rhotrix elements. This setting facilitates a rigorous analysis of rhotrix algebra, specifically identifying the conditions required for invertibility and exploring the structural response of refined neutrosophic elements to basic operations defined for rhotrices. In addition to the invertibility analysis, this study analyzes the index structures of rhotrices of small orders, establishes the general characterization of the rhotrix index set, and reveals an inherent symmetry of the rhotrix structure under the interchange of indices. It also proposes a rhotrix-based representation of binary relations, illustrating the broader applicability of refined neutrosophic rhotrices in modeling relational and algebraic structures. To demonstrate the practical relevance of the developed framework, a refined neutrosophic decision-making model is applied to the evaluation of selected women empowerment schemes. By incorporating real-valued performance measures together with indeterminacy components reflecting expert uncertainty, the proposed model provides a systematic mechanism for multi-criteria assessment and ranking of policy interventions. The results contribute to the advancement of rhotrix algebra in indeterminate environments and highlight its potential applications in mathematical modeling and decision-making.
Gnanachandra et al. (Sun,) studied this question.