From arithmetics of extensional fuzzy numbers to their distances

The notion of the metric space that allows to measure a distance between objects of the given space, has a crucial importance for distinct parts of mathematics, for instance, for the approximation theory, interpolation methods, data analysis, optimization etc. In fuzzy mathematics, the same areas of applications have an analogous importance and thus, not surprisingly measuring the distance between objects possesses a desirable importance. In many cases, e.g., in fuzzy clustering, the use of the standard metric spaces is absolutely sufficient. However, if we deal with vague quantities represented by fuzzy numbers, though the application of a standard metric to fuzzy numbers is mathematically correct, it may lead to counterintuitive and undesirable results. Our investigation constructs the "metric-like" spaces enabling to measure the distance between two fuzzy numbers in a way that is not disconnected from the used arithmetic of fuzzy numbers. Following the analogy from the classical math where the most natural distance between two numbers is the absolute value of their difference, in the case of fuzzy numbers and under the assumption that the distance is connected to the arithmetic, the most natural distance of two fuzzy numbers is the absolute values of their difference too. But then, naturally, the distance should map fuzzy numbers again to fuzzy numbers, not to crisp numbers. This article is a contribution to this area that guides readers from the fundamental notions to the final construction supported by some theoretical results.