Discrete fuzzy numbers, and in particular those defined over a finite chain $L_n = \{0, \ldots, n\}$, have been effectively employed to represent linguistic information within the framework of fuzzy systems. Research on total (admissible) orderings of such types of fuzzy subsets, and specifically those belonging to the set $\mathcal{D}_1^{L_n\rightarrow Y_m}$ consisting of discrete fuzzy numbers $A$ whose support is a closed subinterval of the finite chain $L_n = \{0, 1, \ldots, n\}$ and whose membership values $A(x)$, for $x \in L_n$, belong to the set $Y_m = \{ 0 = y_1 < y_2 < \cdots < y_{m-1} < y_m = 1 \}$, has facilitated the development of new methods for constructing logical connectives, based on a bijective function, called $\textit{pos function}$, that determines the position of each $A \in \mathcal{D}_1^{L_n\rightarrow Y_m}$. For this reason, in this work we revisit the problem by introducing algorithms that exploit the combinatorial structure of total (admissible) orders to compute the $\textit{pos}$ function and its inverse with exactness. The proposed approach achieves a complexity of $\mathcal{O}(n^{2} m \log n)$, which is quadratic in the size of the underlying chain ($n$) and linear in the number of membership levels ($m$). The key point is that the dominant factor is $m$, ensuring scalability with respect to the granularity of membership values. The results demonstrate that this formulation substantially reduces computational cost and enables the efficient implementation of algebraic operations -- such as aggregation and implication -- on the set of discrete fuzzy numbers.

翻译：离散模糊数，特别是定义在有限链 $L_n = \\{0, \\ldots, n\\}$ 上的离散模糊数，已被有效地用于在模糊系统框架内表示语言信息。对此类模糊子集（特别是属于集合 $\\mathcal{D}_1^{L_n\\rightarrow Y_m}$ 的离散模糊数）的全（可容许）序的研究，促进了基于一种称为 $\\textit{pos 函数}$ 的双射函数构建逻辑连接词的新方法的发展。该函数决定了每个 $A \\in \\mathcal{D}_1^{L_n\\rightarrow Y_m}$ 的位置，其中 $A$ 的支撑集是有限链 $L_n = \\{0, 1, \\ldots, n\\}$ 的闭子区间，且其隶属度值 $A(x)$（对于 $x \\in L_n$）属于集合 $Y_m = \\{ 0 = y_1 < y_2 < \\cdots < y_{m-1} < y_m = 1 \\}$。因此，在本工作中，我们通过引入利用全（可容许）序的组合结构来精确计算 $\\textit{pos}$ 函数及其逆的算法，重新审视了该问题。所提出的方法实现了 $\\mathcal{O}(n^{2} m \\log n)$ 的复杂度，该复杂度相对于底层链的大小 ($n$) 是二次的，相对于隶属度级别的数量 ($m$) 是线性的。关键在于主导因子是 $m$，这确保了相对于隶属度值粒度的可扩展性。结果表明，该公式显著降低了计算成本，并使得在离散模糊数集合上高效实现代数运算（如聚合和蕴含）成为可能。