On Fuzzy Implication Functions Based on Admissible Orders on the Set of Discrete Fuzzy Numbers

Research on the construction of logical connectives using total (admissible) orders is a prolific area of study. Using such orders, a new method for constructing implication functions is defined on the set of discrete fuzzy numbers with support of a closed interval of a given finite chain and whose membership values belong to a finite set of fixed values. This method is based on the use of discrete implication functions defined on a finite chain. Furthermore, a bijective correspondence between the set of implication functions on the aforementioned subset of discrete fuzzy numbers and the set of discrete implication functions defined on the discrete chain is shown. Basic properties of these implication functions are thoroughly investigated, concluding that they are preserved under the proposed construction method. This result highlights the robustness and generality of the method, providing a systematic way to extend discrete implication functions to more complex structures while preserving their underlying properties.