The article develops further directions stemming from the arithmetic of extensional fuzzy numbers. It presents the existing knowledge of the relationship between the arithmetic and the proposed orderings of extensional fuzzy numbers—so-called $\mathcal{S}$-orderings—and investigates distinct properties of such orderings. The desirable investigation of the $\mathcal{S}$-orderings of extensional fuzzy numbers is directly used in the concept of $\mathcal{S}$-function—a natural extension of the notion of a function that, in its arguments as well as results, uses extensional fuzzy numbers. One of the immediate subsequent applications is fuzzy interpolation. The article provides readers with the basic fuzzy interpolation method, investigation of its properties and an illustrative experimental example on real data. The goal of the paper is, however, much deeper than presenting a single fuzzy interpolation method. It determines direction to a wide variety of fuzzy interpolation as well as other analytical methods stemming from the concept of $\mathcal{S}$-function and from the arithmetic of extensional fuzzy numbers in general.

中文翻译：

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具有扩展模糊数的模糊插值

本文根据扩展模糊数的算法提出了进一步的发展方向。它提供了关于算术与拟议的扩展模糊数序之间关系的现有知识，即所谓的$\mathcal{小号}$-排序-并研究此类排序的不同属性。理想的调查$\mathcal{小号}$模糊数的阶数直接用于 $\mathcal{小号}$-功能-函数概念的自然扩展，该函数在其参数以及结果中均使用扩展模糊数。直接的后续应用之一是模糊插值。本文为读者提供了基本的模糊插值方法，其性质的研究以及对真实数据的说明性实验示例。但是，本文的目标比提出一种模糊插值方法要深得多。它确定了各种模糊插值的方向以及源于“模糊插值”概念的其他分析方法。$\mathcal{小号}$函数和一般的扩展模糊数算法。