## 5 Conclusions  
This paper introduces a redefinition of Ordered Fuzzy Numbers (OFNs) through a novel representation based on vector pairs, termed Vectorially Ordered Fuzzy Numbers (vOFNs). In this approach, the classical view of an OFN as an ordered pair of functions is replaced by an ordered pair of vectors. This shift addresses several limitations of previous fuzzy number formalisms, such as the LR notation by [PERSON], which restricts the edges to rectilinear shapes and precludes curvilinear transformations. The proposed vOFN formulation constitutes a significant departure from the function-based definition of OFNs by offering a fully vectorial structure. Unlike conventional LR-type fuzzy numbers [6], which enforce parametric symmetry and require defuzzification for comparative reasoning, vOFNs allow direct vector-based operations and comparisons, preserving both directionality and shape semantics without approximation. This contributes to improved interpretability and reduced computational burden in practical systems. At the same time, the vOFN framework retains the beneficial properties of [PERSON]'s original OFN concept. It forms a complete ring structure, with neutral elements for addition,  
Figure 13: Normalization of fuzzy numbers \(A_{1}=[1,3,3,2]\) and \(A_{2}=[3,4,1,2]\). subtraction, multiplication, and division. Operations are well-defined across a broad class of fuzzy numbers, including triangular and trapezoidal forms and singletons. Notably, vOFNs maintain the directional ordering property, enhancing the expressive capability of fuzzy numbers by encoding magnitude, uncertainty, and directional trends simultaneously. Moreover, a new vector-based similarity metric has been introduced to evaluate the closeness of fuzzy numbers without requiring the defuzzification or centroid approximation. This metric preserves both the spatial structure and the directional profile of fuzzy numbers, and it serves as a foundation for comparison-based reasoning in intelligent systems. To illustrate practical integration, a simplified implementation scenario is outlined for incorporating vOFN arithmetic into large language models (LLMs). Specifically, fuzzy arithmetic based on vOFNs can be embedded into the attention or weighting mechanisms of LLMs to represent uncertainty in token or context relevance scores. For instance, a fuzzy-weighted attention mechanism may use vOFNs to evaluate ambiguous or under-specified user input by modeling lexical or semantic uncertainty more effectively than crisp probabilities. This offers improved adaptability and interpretability in applications such as fuzzy querying, uncertain context tracking, or intent classification in generative agents. Preliminary implementations using Python-based libraries (e.g., PyTorch and TensorFlow) can incorporate vOFN arithmetic via custom differentiable layers handling fuzzy vector operations, allowing smooth integration into existing transformer-based architectures. Such applications showcase how vOFNs can enhance generative AI systems by enabling flexible, uncertainty-aware reasoning in a mathematically grounded manner. In contrast to other fuzzy models such as intuitionistic fuzzy sets (IFS) [48] and type-2 fuzzy systems [49], which often incur high computational costs due to their layered or interval-based architectures, the vectorial representation of vOFNs ensures linear time operations and easier implementation in differentiable programming environments. A comparative analysis provided in Sections 3.1-3.3 confirms that vOFNs outperform traditional OFNs and LR numbers in terms of generality, algebraic closure, and suitability for algorithmic use. The vOFN arithmetic is versatile and could find a variety of applications, including in the realm of Artificial Intelligence (AI). One such example, discussed above, is the modeling of context-sensitive behaviors in LLMs.