MI-algebras, extensional fuzzy numbers, similarity relation, extensionality, orderings, approximate reasoning

The article stems from distinct studies on arithmetics of fuzzy numbers, orderings of fuzzy numbers, and metrics on fuzzy numbers. Trying to capture the existing knowledge in the mentioned areas and putting them together, we motivate the construction of metric-like spaces on fuzzy numbers by desirable connection to their arithmetics. The desirable "metric" should be mapping pairs of fuzzy numbers again to fuzzy numbers and thus, reflecting the vagueness of operation on fuzzy numbers. This leads to developing all such areas under the joint umbrella and connecting such basic notions as orderings of fuzzy numbers to arithmetics and elementary "metrics" such as the absolute value of the difference of two fuzzy numbers. This article focuses mainly on the orderings and investigation of the preservation of their most natural properties. However, links to further studies going towards applications are also foreshadowed and referred to.