An Understandable Way to Extend the Ordinary Linear Order on Real Numbers to a Linear Order on Interval Numbers

Inspired by reflection transformations and rotation transformations in plane geometry and dictionary orders in lattice theory, in this article, we explore, through exposing the possible motivation of definition of a defined linear order ≤ on R (the set of all interval numbers), how to find an easy-to-understand way to extend the ordinary linear order on R (the set of all real numbers) to a linear order on R. Theory and application aspects are also studied. For an arbitrary cardinal number κ (κ ≠ 0), we define seven operations and the point-wise order induced by ≤ on R ^κ (the set of all sequences indexed by κ and consisting of interval numbers). It is found that ( R,\≤), much like R (with the ordinary linear order), has some desirable properties; particularly, ≤ can be used to describe the corrected degree of possibility of an interval number is smaller than another. Generalizing the discovery process of ≤, we provide an understandable way to extend the ordinary linear order on R to a linear order on R (even on the plane R ² ), which meets some specific requirement, and we give a clear description on the relation between the admissible orders and those we propose. Those urge us to take a look at the possible applications of ≤ (a delegation of linear orders on R). Precisely, to propose an ordering method (based on this linear order and the matched metric and weighted averaging aggregation operator), which can be used to deal with very complicated generalized interval-valued preference hesitant relations in group decision-making problems.