On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers

Source abstract

This paper is devoted to the investigation of ideal-type level-wise convergence for sequences of fuzzy numbers. By using $\alpha$-level sets, we define level-wise $\mathcal I$-convergence, uniformly level-wise $\mathcal I$-convergence, and almost everywhere level-wise $\mathcal I$-convergence. A characterization of level-wise $\mathcal I$-convergence is obtained via the endpoint functions of the $\alpha$-cuts, and uniqueness of the limit is established. It is shown that uniformly level-wise $\mathcal I$-convergence implies level-wise $\mathcal I$-convergence, whereas the converse fails in general. Furthermore, we prove that level-wise $\mathcal I$-convergence yields almost everywhere level-wise $\mathcal I$-convergence, whereas the converse implication does not hold in general. Several examples are included to illustrate the strictness of these implications.