Generated admissible orders for intervals by matrices and continuous functions

Abstract

In this paper, we systematically study the algebraic structures of the spaces L([0,1]), encompassing all closed subintervals of [0,1], under the generated admissible orders. We first prove that the admissible order on L([0,1]) generated by a non-degenerate matrix must be the form of two weighted averaging operators. As a corollary, we deduce that each admissible order on L([0,1]) generated by a non-degenerate matrix and the standard order ≤ on [0,1] are not isomorphic. Furthermore, we show that each admissible order on L([0,1]) derived from two continuous mappings and the standard order ≤ on [0,1] are not isomorphic, partially answering a conjecture proposed by Santana et al. (2020) [38]. Besides, we prove that L([0,1]) is a complete lattice under the admissible order generated by two continuous mappings. This is the first result regarding the completeness of L([0,1]). Finally, we apply the admissible orders to solve a minimal path problem within the context of interval-valued fuzzy weighted graph. The above results theoretically refine the study of the classification, non-isomorphism, and completeness of admissible orders, while expanding the scope of interval-valued fuzzy sets in practical applications.